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HYDRAULICS (Gr. ὕδωρ, water, and αὐλός, a pipe), the branch of engineering science which deals with the practical applications of the laws of hydromechanics.

I. THE DATA OF HYDRAULICS[1]

§ 1. Properties of Fluids.—The fluids to which the laws of practical hydraulics relate are substances the parts of which possess very great mobility, or which offer a very small resistance to distortion independently of inertia. Under the general heading Hydromechanics a fluid is defined to be a substance which yields continually to the slightest tangential stress, and hence in a fluid at rest there can be no tangential stress. But, further, in fluids such as water, air, steam, &c., to which the present division of the article relates, the tangential stresses that are called into action between contiguous portions during distortion or change of figure are always small compared with the weight, inertia, pressure, &c., which produce the visible motions it is the object of hydraulics to estimate. On the other hand, while a fluid passes easily from one form to another, it opposes considerable resistance to change of volume.

It is easily deduced from the absence or smallness of the tangential stress that contiguous portions of fluid act on each other with a pressure which is exactly or very nearly normal to the interface which separates them. The stress must be a pressure, not a tension, or the parts would separate. Further, at any point in a fluid the pressure in all directions must be the same; or, in other words, the pressure on any small element of surface is independent of the orientation of the surface.

§ 2. Fluids are divided into liquids, or incompressible fluids, and gases, or compressible fluids. Very great changes of pressure change the volume of liquids only by a small amount, and if the pressure on them is reduced to zero they do not sensibly dilate. In gases or compressible fluids the volume alters sensibly for small changes of pressure, and if the pressure is indefinitely diminished they dilate without limit.

In ordinary hydraulics, liquids are treated as absolutely incompressible. In dealing with gases the changes of volume which accompany changes of pressure must be taken into account.

§ 3. Viscous fluids are those in which change of form under a continued stress proceeds gradually and increases indefinitely. A very viscous fluid opposes great resistance to change of form in a short time, and yet may be deformed considerably by a small stress acting for a long period. A block of pitch is more easily splintered than indented by a hammer, but under the action of the mere weight of its parts acting for a long enough time it flattens out and flows like a liquid.

EB1911 Hydraulics - Fig. 1.jpg
Fig. 1.

All actual fluids are viscous. They oppose a resistance to the relative motion of their parts. This resistance diminishes with the velocity of the relative motion, and becomes zero in a fluid the parts of which are relatively at rest. When the relative motion of different parts of a fluid is small, the viscosity may be neglected without introducing important errors. On the other hand, where there is considerable relative motion, the viscosity may be expected to have an influence too great to be neglected.

Measurement of Viscosity. Coefficient of Viscosity.—Suppose the plane ab, fig. 1 of area ω, to move with the velocity V relatively to the surface cd and parallel to it. Let the space between be filled with liquid. The layers of liquid in contact with ab and cd adhere to them. The intermediate layers all offering an equal resistance to shearing or distortion, the rectangle of fluid abcd will take the form of the parallelogram abcd. Further, the resistance to the motion of ab may be expressed in the form

R = κωV,
(1)

where κ is a coefficient the nature of which remains to be determined.

If we suppose the liquid between ab and cd divided into layers as shown in fig. 2, it will be clear that the stress R acts, at each dividing face, forwards in the direction of motion if we consider the upper layer, backwards if we consider the lower layer. Now suppose the original thickness of the layer T increased to nT; if the bounding plane in its new position has the velocity nV, the shearing at each dividing face will be exactly the same as before, and the resistance must therefore be the same. Hence,

R = κω (nV).
(2)

But equations (1) and (2) may both be expressed in one equation if κ and κ′ are replaced by a constant varying inversely as the thickness of the layer. Putting κ = μ/T, κ′ = μ/nT,

R = μωV/T;

or, for an indefinitely thin layer,

R = μωdV/dt,
(3)

an expression first proposed by L. M. H. Navier. The coefficient μ is termed the coefficient of viscosity.

According to J. Clerk Maxwell, the value of μ for air at θ° Fahr. in pounds, when the velocities are expressed in feet per second, is

μ = 0.000 000 025 6 (461° + θ);

that is, the coefficient of viscosity is proportional to the absolute temperature and independent of the pressure.

The value of μ for water at 77° Fahr. is, according to H. von Helmholtz and G. Piotrowski,

μ = 0.000 018 8,

the units being the same as before. For water μ decreases rapidly with increase of temperature.

EB1911 Hydraulics - Fig. 2.jpg
Fig. 2.

§ 4. When a fluid flows in a very regular manner, as for instance when it flows in a capillary tube, the velocities vary gradually at any moment from one point of the fluid to a neighbouring point. The layer adjacent to the sides of the tube adheres to it and is at rest. The layers more interior than this slide on each other. But the resistance developed by these regular movements is very small. If in large pipes and open channels there were a similar regularity of movement, the neighbouring filaments would acquire, especially near the sides, very great relative velocities. V. J. Boussinesq has shown that the central filament in a semicircular canal of 1 metre radius, and inclined at a slope of only 0.0001, would have a velocity of 187 metres per second,[2] the layer next the boundary remaining at rest. But before such a difference of velocity can arise, the motion of the fluid becomes much more complicated. Volumes of fluid are detached continually from the boundaries, and, revolving, form eddies traversing the fluid in all directions, and sliding with finite relative velocities against those surrounding them. These slidings develop resistances incomparably greater than the viscous resistance due to movements varying continuously from point to point. The movements which produce the phenomena commonly ascribed to fluid friction must be regarded as rapidly or even suddenly varying from one point to another. The internal resistances to the motion of the fluid do not depend merely on the general velocities of translation at different points of the fluid (or what Boussinesq terms the mean local velocities), but rather on the intensity at each point of the eddying agitation. The problems of hydraulics are therefore much more complicated than problems in which a regular motion of the fluid is assumed, hindered by the viscosity of the fluid.

Relation of Pressure, Density, and Temperature of Liquids

§ 5. Units of Volume.—In practical calculations the cubic foot and gallon are largely used, and in metric countries the litre and cubic metre (= 1000 litres). The imperial gallon is now exclusively used in England, but the United States have retained the old English wine gallon.

1 cub. ft.  = 6.236 imp. gallons  = 7.481 U.S. gallons.
1 imp. gallon  = 0.1605 cub. ft.  = 1.200 U.S. gallons.
1 U.S. gallon  = 0.1337 cub. ft.  = 0.8333 imp. gallon.
1 litre  = 0.2201 imp. gallon  = 0.2641 U.S. gallon.

Density of Water.—Water at 53° F. and ordinary pressure contains 62.4 ℔ per cub. ft., or 10 ℔ per imperial gallon at 62° F. The litre contains one kilogram of water at 4° C. or 1000 kilograms per cubic metre. River and spring water is not sensibly denser than pure water. But average sea water weighs 64 ℔ per cub. ft. at 53° F. The weight of water per cubic unit will be denoted by G. Ice free from air weighs 57.28 ℔ per cub. ft. (Leduc).

§ 6. Compressibility of Liquids.—The most accurate experiments show that liquids are sensibly compressed by very great pressures, and that up to a pressure of 65 atmospheres, or about 1000 ℔ per sq. in., the compression is proportional to the pressure. The chief results of experiment are given in the following table. Let V1 be the volume of a liquid in cubic feet under a pressure p1 ℔ per sq. ft., and V2 its volume under a pressure p2. Then the cubical compression is (V2 − V1)/V1, and the ratio of the increase of pressure p2p1 to the cubical compression is sensibly constant. That is, k = (p2p1)V1/(V2 − V1) is constant. This constant is termed the elasticity of volume. With the notation of the differential calculus,

k = dp / ( dV ) = − V dp .
V dV

Elasticity of Volume of Liquids.

  Canton. Oersted. Colladon
and Sturm.
Regnault.
Water 45,990,000 45,900,000 42,660,000 44,000,000
Sea water 52,900,000 ·· ·· ··
Mercury 705,300,000 ·· 626,100,000 604,500,000
Oil 44,090,000 ·· ·· ··
Alcohol 32,060,000 ·· 23,100,000 ··

According to the experiments of Grassi, the compressibility of water diminishes as the temperature increases, while that of ether, alcohol and chloroform is increased.

§ 7. Change of Volume and Density of Water with Change of Temperature.—Although the change of volume of water with change of temperature is so small that it may generally be neglected in ordinary hydraulic calculations, yet it should be noted that there is a change of volume which should be allowed for in very exact calculations. The values of ρ in the following short table, which gives data enough for hydraulic purposes, are taken from Professor Everett’s System of Units.

Density of Water at Different Temperatures.

Temperature. ρ
Density of
Water.
G
Weight of
1 cub. ft.
in ℔.
 Cent.  Fahr.
0 32.0 .999884 62.417
1 33.8 .999941 62.420
2 35.6 .999982 62.423
3 37.4  1.000004 62.424
4 39.2 1.000013 62.425
5 41.0 1.000003 62.424
6 42.8 .999983 62.423
7 44.6 .999946 62.421
8 46.4 .999899 62.418
9 48.2 .999837 62.414
10 50.0 .999760 62.409
11 51.8 .999668 62.403
12 53.6 .999562 62.397
13 55.4 .999443 62.389
14 57.2 .999312 62.381
15 59.0 .999173 62.373
16 60.8 .999015 62.363
17 62.6 .998854 62.353
18 64.4 .998667 62.341
19 66.2 .998473 62.329
20 68.0 .998272 62.316
22 71.6 .997839 62.289
24 75.2 .997380 62.261
26 78.8 .996879 62.229
28 82.4 .996344 62.196
30 86   .995778 62.161
35 95   .99469  62.093
40 104   .99236  61.947
45 113   .99038  61.823
50 122   .98821  61.688
55 131   .98583  61.540
60 140   .98339  61.387
65 149   .98075  61.222
70 158   .97795  61.048
75 167   .97499  60.863
80 176   .97195  60.674
85 185   .96880  60.477
90 194   .96557  60.275
100  212   .95866  59.844

The weight per cubic foot has been calculated from the values of ρ, on the assumption that 1 cub. ft. of water at 39.2° Fahr. is 62.425 ℔. For ordinary calculations in hydraulics, the density of water (which will in future be designated by the symbol G) will be taken at 62.4 ℔ per cub. ft., which is its density at 53° Fahr. It may be noted also that ice at 32° Fahr. contains 57.3 ℔ per cub. ft. The values of ρ are the densities in grammes per cubic centimetre.

§ 8. Pressure Column. Free Surface Level.—Suppose a small vertical pipe introduced into a liquid at any point P (fig. 3). Then the liquid will rise in the pipe to a level OO, such that the pressure due to the column in the pipe exactly balances the pressure on its mouth. If the fluid is in motion the mouth of the pipe must be supposed accurately parallel to the direction of motion, or the impact of the liquid at the mouth of the pipe will have an influence on the height of the column. If this condition is complied with, the height h of the column is a measure of the pressure at the point P. Let ω be the area of section of the pipe, h the height of the pressure column, p the intensity of pressure at P; then

pω = Ghω ℔,
p/G = h;

that is, h is the height due to the pressure at p. The level OO will be termed the free surface level corresponding to the pressure at P.

Relation of Pressure, Temperature, and Density of Gases

EB1911 Hydraulics - Fig. 3.jpg
Fig. 3.

§ 9. Relation of Pressure, Volume, Temperature and Density in Compressible Fluids.—Certain problems on the flow of air and steam are so similar to those relating to the flow of water that they are conveniently treated together. It is necessary, therefore, to state as briefly as possible the properties of compressible fluids so far as knowledge of them is requisite in the solution of these problems. Air may be taken as a type of these fluids, and the numerical data here given will relate to air.

Relation of Pressure and Volume at Constant Temperature.—At constant temperature the product of the pressure p and volume V of a given quantity of air is a constant (Boyle’s law).

Let p0 be mean atmospheric pressure (2116.8 ℔ per sq. ft.), V0 the volume of 1 ℔ of air at 32° Fahr. under the pressure p0. Then

p0V0 = 26214.
(1)

If G0 is the weight per cubic foot of air in the same conditions,

G0 = 1/V0 = 2116.8/26214 = .08075.
(2)

For any other pressure p, at which the volume of 1 ℔ is V and the weight per cubic foot is G, the temperature being 32° Fahr.,

pV = p/G = 26214; or G = p/26214.
(3)

Change of Pressure or Volume by Change of Temperature.—Let p0, V0, G0, as before be the pressure, the volume of a pound in cubic feet, and the weight of a cubic foot in pounds, at 32° Fahr. Let p, V, G be the same quantities at a temperature t (measured strictly by the air thermometer, the degrees of which differ a little from those of a mercurial thermometer). Then, by experiment,

pV = p0V0 (460.6 + t) / (460.6 + 32) = p0V0τ/τ0,
(4)

where τ, τ0 are the temperatures t and 32° reckoned from the absolute zero, which is −460.6° Fahr.;

p/G = p0τ/G0τ0;
(4a)
G = pτ0G0/p0τ.
(5)

If p0 = 2116.8, G0 = .08075, τ0 = 460.6 + 32 = 492.6, then

p/G = 53.2τ.
(5a)

Or quite generally p/G = Rτ for all gases, if R is a constant varying inversely as the density of the gas at 32° F. For steam R = 85.5.

II. KINEMATICS OF FLUIDS

§ 10. Moving fluids as commonly observed are conveniently classified thus:

(1) Streams are moving masses of indefinite length, completely or incompletely bounded laterally by solid boundaries. When the solid boundaries are complete, the flow is said to take place in a pipe. When the solid boundary is incomplete and leaves the upper surface of the fluid free, it is termed a stream bed or channel or canal.

(2) A stream bounded laterally by differently moving fluid of the same kind is termed a current.

(3) A jet is a stream bounded by fluid of a different kind.

(4) An eddy, vortex or whirlpool is a mass of fluid the particles of which are moving circularly or spirally.

(5) In a stream we may often regard the particles as flowing along definite paths in space. A chain of particles following each other along such a constant path may be termed a fluid filament or elementary stream.

§ 11. Steady and Unsteady, Uniform and Varying, Motion.—There are two quite distinct ways of treating hydrodynamical questions. We may either fix attention on a given mass of fluid and consider its changes of position and energy under the action of the stresses to which it is subjected, or we may have regard to a given fixed portion of space, and consider the volume and energy of the fluid entering and leaving that space.

EB1911 Hydraulics - Fig. 4.jpg
Fig. 4.

If, in following a given path ab (fig. 4), a mass of water a has a constant velocity, the motion is said to be uniform. The kinetic energy of the mass a remains unchanged. If the velocity varies from point to point of the path, the motion is called varying motion. If at a given point a in space, the particles of water always arrive with the same velocity and in the same direction, during any given time, then the motion is termed steady motion. On the contrary, if at the point a the velocity or direction varies from moment to moment the motion is termed unsteady. A river which excavates its own bed is in unsteady motion so long as the slope and form of the bed is changing. It, however, tends always towards a condition in which the bed ceases to change, and it is then said to have reached a condition of permanent regime. No river probably is in absolutely permanent regime, except perhaps in rocky channels. In other cases the bed is scoured more or less during the rise of a flood, and silted again during the subsidence of the flood. But while many streams of a torrential character change the condition of their bed often and to a large extent, in others the changes are comparatively small and not easily observed.

As a stream approaches a condition of steady motion, its regime becomes permanent. Hence steady motion and permanent regime are sometimes used as meaning the same thing. The one, however, is a definite term applicable to the motion of the water, the other a less definite term applicable in strictness only to the condition of the stream bed.

§ 12. Theoretical Notions on the Motion of Water.—The actual motion of the particles of water is in most cases very complex. To simplify hydrodynamic problems, simpler modes of motion are assumed, and the results of theory so obtained are compared experimentally with the actual motions.

EB1911 Hydraulics - Fig. 5.jpg
Fig. 5.

Motion in Plane Layers.—The simplest kind of motion in a stream is one in which the particles initially situated in any plane cross section of the stream continue to be found in plane cross sections during the subsequent motion. Thus, if the particles in a thin plane layer ab (fig. 5) are found again in a thin plane layer ab′ after any interval of time, the motion is said to be motion in plane layers. In such motion the internal work in deforming the layer may usually be disregarded, and the resistance to the motion is confined to the circumference.

Laminar Motion.—In the case of streams having solid boundaries, it is observed that the central parts move faster than the lateral parts. To take account of these differences of velocity, the stream may be conceived to be divided into thin laminae, having cross sections somewhat similar to the solid boundary of the stream, and sliding on each other. The different laminae can then be treated as having differing velocities according to any law either observed or deduced from their mutual friction. A much closer approximation to the real motion of ordinary streams is thus obtained.

Stream Line Motion.—In the preceding hypothesis, all the particles in each lamina have the same velocity at any given cross section of the stream. If this assumption is abandoned, the cross section of the stream must be supposed divided into indefinitely small areas, each representing the section of a fluid filament. Then these filaments may have any law of variation of velocity assigned to them. If the motion is steady motion these fluid filaments (or as they are then termed stream lines) will have fixed positions in space.

EB1911 Hydraulics - Fig. 6.jpg
Fig. 6.

Periodic Unsteady Motion.—In ordinary streams with rough boundaries, it is observed that at any given point the velocity varies from moment to moment in magnitude and direction, but that the average velocity for a sensible period (say for 5 or 10 minutes) varies very little either in magnitude or velocity. It has hence been conceived that the variations of direction and magnitude of the velocity are periodic, and that, if for each point of the stream the mean velocity and direction of motion were substituted for the actual more or less varying motions, the motion of the stream might be treated as steady stream line or steady laminar motion.

§ 13. Volume of Flow.—Let A (fig. 6) be any ideal plane surface, of area ω, in a stream, normal to the direction of motion, and let V be the velocity of the fluid. Then the volume flowing through the surface A in unit time is

Q = ωV.
(1)

Thus, if the motion is rectilinear, all the particles at any instant in the surface A will be found after one second in a similar surface A′, at a distance V, and as each particle is followed by a continuous thread of other particles, the volume of flow is the right prism AA′ having a base ω and length V.

If the direction of motion makes an angle θ with the normal to the surface, the volume of flow is represented by an oblique prism AA′ (fig. 7), and in that case

Q = ωV cos θ.
EB1911 Hydraulics - Fig. 7.jpg
Fig. 7.

If the velocity varies at different points of the surface, let the surface be divided into very small portions, for each of which the velocity may be regarded as constant. If dω is the area and v, or v cos θ, the normal velocity for this element of the surface, the volume of flow is

Q = ∫ v dω, or ∫ v cos θ dω,

as the case may be.

§ 14. Principle of Continuity.—If we consider any completely bounded fixed space in a moving liquid initially and finally filled continuously with liquid, the inflow must be equal to the outflow. Expressing the inflow with a positive and the outflow with a negative sign, and estimating the volume of flow Q for all the boundaries,

ΣQ = 0.

In general the space will remain filled with fluid if the pressure at every point remains positive. There will be a break of continuity, if at any point the pressure becomes negative, indicating that the stress at that point is tensile. In the case of ordinary water this statement requires modification. Water contains a variable amount of air in solution, often about one-twentieth of its volume. This air is disengaged and breaks the continuity of the liquid, if the pressure falls below a point corresponding to its tension. It is for this reason that pumps will not draw water to the full height due to atmospheric pressure.

Application of the Principle of Continuity to the case of a Stream.—If A1, A2 are the areas of two normal cross sections of a stream, and V1, V2 are the velocities of the stream at those sections, then from the principle of continuity,

V1A1 = V2A2;
V1/V2 = A2/A1
(2)

that is, the normal velocities are inversely as the areas of the cross sections. This is true of the mean velocities, if at each section the velocity of the stream varies. In a river of varying slope the velocity varies with the slope. It is easy therefore to see that in parts of large cross section the slope is smaller than in parts of small cross section.

If we conceive a space in a liquid bounded by normal sections at A1, A2 and between A1, A2 by stream lines (fig. 8), then, as there is no flow across the stream lines,

V1/V2 = A2/A1,

as in a stream with rigid boundaries.

EB1911 Hydraulics - Fig. 8.jpg
Fig. 8.

In the case of compressible fluids the variation of volume due to the difference of pressure at the two sections must be taken into account. If the motion is steady the weight of fluid between two cross sections of a stream must remain constant. Hence the weight flowing in must be the same as the weight flowing out. Let p1, p2 be the pressures, v1, v2 the velocities, G1, G2 the weight per cubic foot of fluid, at cross sections of a stream of areas A1, A2. The volumes of inflow and outflow are

A1v1 and A2v2,

and, if the weights of these are the same,

G1A1v1 = G2A2v2;

and hence, from (5a) § 9, if the temperature is constant,

p1A1v1 = p2A2v2.
(3)

§ 1 Stream Lines.-The characteristic of a perfect fluid, that is, a flui free from viscosity, is that the pressure between any two parts into which it is divided by a plane must be normal to the plane. One consequence of this is that the particles can have no rotation impressed upon them, and the motion of such a fluid is irrotational. A stream line is the line, straight or curved, traced by a particle in a current of fluid in irrotational movement. In a steady current "'

r, .

“ ~ 2

e

I <

 /{

H U FIG.9.

each stream line preserves its figure and position unchanged, and marks the track of a stream of particles forming a fluid filament or elementary stream. A current in steady irrotational movement may be conceived to be divided by insensibly thin partitions following the course of the stream lines into a number of elementary streams. If the positions of these partitions are so adjusted that the volumes of flow in all the elementary streams are equal, they represent to the mind the velocity as well as the direction of motion of the particles in different parts of the current, for the velocities M V ' U

fl K , ff '<' wQ (U

< i f: 1 K

if FIiG- 10. Fig. 11. Fig. 12.

are inversely proportional to the cross sections of the elementary streams. No actual fluid is devoid of viscosity, and the effect of viscosity is to render the motion of a fluid sinuous, or rotational or eddying under most ordinary conditions. At very low velocities in a tube of moderate size the motion of water may be nearly pure stream line motion. But at some velocity, smaller as the diameter of the tube is greater, the motion suddenly becomes tumultuous. The laws of simple stream line motion have hitherto been investigated theoretically, and from mathematical difficulties have only been determined for certain simple cases. Professor H. S. Hele Shaw has found means of exhibiting stream line motion in a number of very interesting cases experimentally. Generally in these experiments a thin sheet of fluid is caused to flow between two parallel plates of glass. In the earlier experiments streams of very small air ¢, , bubbles introduced into the water current rendered visible the motions of the water. By the use of a lantern the image of a portion of the current can be shown on a screen or photographed. In later experiments streams of

coloured liquid at regular distances were intro) duced into the sheet and these much more clearly marked out the forms of the stream lines. With a fluid sheet 0-02 in. thick, the I, stream lines were found to be stable at almost any required velocity. For certain simple cases Professor Hele Shaw has shown that the experimental stream lines of a viscous fluid are so far as can be measured identical with the calculated stream lines of a perfect fluid. Sir G. G. Stokes pointed out that in this case, either from the thinness of the stream between its glass walls, or the slowness of the motion, or the high viscosity of the liquid, or from Z1 combination of all these, the flow is regular, and the effects of inertia disappear, the viscosity dominating everything. Glycerine gives the stream lines very satisfactorily. FIG. 9 shows the stream lines of a sheet of fluid passing a fairly FIG. 13.

shipshape body such as a screw shaft strut. The arrow shows the direction of motion of the fluid. Fig. IO shows the stream lines for a very thin glycerin sheet passing a non-shipshape body, the stream lines being practically perfect. Fig. II shows one of the earlier air-bubble experiments with a thicker sheet of water. In this case the stream lines break up behind the obstruction, forming an eddying wake. Fig. 12 shows the stream lines of a fluid passing a sudden contraction or sudden enlargement of a pipe. Lastly, fig. 13 shows the stream lines of a current passing an oblique plane. H. S. Hele Shaw, “ Experiments on the Nature of the Surface Resistance in Pipes and on Ships, ” Trans. Inst. Naval Arch. (1897). “ Investigation of Stream Line Motion under certain Experimental Conditions, ” Trans. Inst. Naval/lrch. (1898); “ Stream Line Motion of a Viscous Fluid, ” Report of British Association (1898). III. PHENOMENA OF THE DISCHARGE OF LIQUIDS FROM ORIFICES AS ASCERTAINABLE BY EXPERIMENTS § 16. when a liquid issues vertically from a small orifice, it forms a Jet which rises nearly to the level of the free surface of the liquid in the vessel from which

it flows. The difference

of level hr (fig. 14) is

so small that it may be

hr

%, fi -

at once suspected to be —— - in

due either to air resistance by, ,, ,, I 2 on the surface of the jet, Wi l ily I Q or to the viscosity of the H, J 1 if g 5 liquid or to friction again st, I

the sides of the orifice. . I

Neglecting for the moment ** ""“' ' 1 3 this small quantity, we i 5

may infer, from the eleva- I ll

tion of the jet, that each ~ he 2

molecule on leaving the 3

orifice possessed the velo- I 1

city required to lift it i V I

against gravity to the } {

height h. From ordinary 1 1

dynamics, the relation —.l— .l.

between the velocity and "

height of projection is

given by the fguation

v=/zgh. (1)

As this velocity is nearly #ff

reached in the flow from

well-formed orifices, it is

FIG. 14.

sometimes called the theoretical velocity of discharge. This relation was first obtained by Torricelli.

If the orifice is of a suitable conoidal form, the water issues in filaments normal to the plane of the orifice. Let w be the area of the orifice, then the discharge per second must be, from eq. (1), Q =w1J =w/ 2gh nearly. (2)

This is sometimes quite improperly called the theoretical discharge for any kind of orifice. Except for a well-formed conoidal orifice the result is not a proximate even, so that if it is supposed to be based on a theory the theory is a false one. Use of the term Head in Hydraulics.-The term head is an old millwri ht's term, and meant primarily the height through which a mass of water descended in actuating a hydraulic machine. Since the water in fig. 14 descends through a height h to the orifice, we may say there are h ft. of head above the orifice. Still more generally any mass of liquid h ft. above a horizontal plane may be said to have h ft. of elevation head relatively to that datum plane. Further, since the pressure lp at the orifice which produces outflow is connected with h by the re ation p/G=h, the quantity p/G may be termed the pressure head at the orifice. Lastly, the velocity 1/ is connected with h by the relation 11”/2g=h, so that 712/2g may be termed the head due to the velocity 11.

§ I 7. Coeficients of Velocity and Resistance.-As the actual velocity of discharge differs from ~/'Eh by a small quantity, let the actual velocity

=°v.. =v., / 2gh, (3)

where e., is a coefficient to be determined by experiment, called the coejicient of velocity. This coefficient is found to be tolerably constant for different heads with well-formed simple orifices, and it very often has the value 0-97.

The difference between the velocity of discharge and the velocity due to the head may be reckoned in another way. The total hei ht h causing outflow consists of two parts—one part h. expended effectively in producing the velocity of outflow, another h, in overcoming the resistances due to viscosity and friction. Let hr-cfhe.

where c, is a coefficient determined by experiment, and called the coejieient of resistance of the orifice. It is tolerably constant for different heads with well-formed orifices. Then va=/§ l'e=/l2gh/(l'i"Cr>l The relation between cv and c, for any orifice is easily found:- ”“ =””'/?@=~'f izff/3; iii” < > 0, = I 1 c, . 5 C, =I/L', ,2°-I. (5a) Thus if c., =O'97, then c, =o-0628. That is, for such an orifice about 6§ ;¥, of the head is expended in overcoming frictional resistances t w. 0 Cziejhcient of Contraction-Sharp-edged Orzjfices in Plane Surfaces.-When a jet issues from an aperture in a. vessel, it may either spring T. ;/, <

Y - ';; "" s

/ it Qi§ ;;§ 0 b C FIG. 15. clear from the inner edge of the orifice as at a or b (fig. 15), or it may adhere to the sides of the orifice as at c. The former condition will be found if the orifice is bevelled outwards as at a, so as to be sharp edged, and it will also occur generally for a prismatic aperture like b, provided the thickness of the plate in which the aperture is formed is less than the diameter of the jet. But if the thickness > the orifice, and t the time in which a particle moves from O to A, then x =1J., t; y = ¥gt2. Eliminating t, 2 Va = xl (§ x2/2y)Then cv =°v.»/~/ (2gh) =/ <x2/4yh>-In the case of large orifices such as weirs, the velocity can be directly determined by using a Pitot tube (§ 144). The coefficient of discharge, which for practical purposes is the most important of measurement of the flow from the given orifice in a suitable time. If Q is the discharge measured in the 5 tank per second, I 0 then C =Q/uv <2gh>. of ¢ Measurements of ~ 1 this kind though I I simple in principle are not free from some practical difficulties, and require m uch care. In fig. 18 is shown an arrangement of measuring tank. The orifice is fixed in the wall of the channel BB, or into the measuring tank. There is a short trough on rollers C which when run under the jet directs the discharge into the tank, and when run back again allows the discharge to drop the three coefficients, is best determined by tank

. T, . ~r*;

2 - £ 11 X F, ., ., I . gy A gf/1 FIG. 17. cistern A and discharges either into the waste is greater the condition shown 3, t%/gill Qirlcula A C en the iscliarge occurs T, . as at a or b. the filaments con- ~ ~- ~ - " g - L verging towards the orifice, . ' ' L '/f B continue to 'converge be?/oigd T 'ET 2 'rg /% 'j %l' lf., SOl13.l2tCS€Ct1Ol'1O te .1 '. “" """°""""' " jet where the filaments have 'J if ' E become parallel is smaller than ' ips! the section of the orifice. The J; ' ~inertia of the filaments opposes - 5 / ' % WVV, ,, ,, Y, , ,> Y , sudden change of direction 5” , , WAN W W, , ' "" of motion at the edge of the P ff?-'=:4;-. f-. 'ef F 4-= J;:, ?? ~- ¢z}'i " ' »::- oriflce, and the convergence continues for a distance of about half the diameter of the orifice beyond it. Let w be the area of the orifice, and C¢w the area of the jet at the point where convergence ceases; then 0, is a coefficient to be determined experimentally for each kind of orifice, called the coejicient of contraction. When the orifice is a sharp-edged orifice in a plane surface, the value of cc is on the average o~64, or the section of the jet is very nearly five-eighths of the area of the orifice. Coejicient of Di5charge.»~In applying the general formula Q=wv to a stream, it is assumed that the filaments have a common velocity 11 normal to the section tu. But if Xml '/jf '/M the jet contracts, it is at the confy /' Q, tracted section of the jet that 5 6 / the direction of motion is normal j, i M'//V to a transverse section of the jet. Hence the actual discharge / W // when contraction occurs is Q..=c, v><c¢w=c¢cu /(zgh), or simplyQif c=c, , c6, ai) where c is called the coejicient 2 ~;¢, »:, H ; 06 dréchaqge. 'lhiés for a sharp-lli|Plzl. S1l!< ~ §§ .§ e ge pane Ori Ce c=()~97)(

. -~ 

§ W 0-64-0-62.

I § 18. E:cper1mentalDetermznajf|,), ,

.W tion of c, ,, c¢, and c.-The coefficient of contraction c, is FIG 16 directly determined by measur-For this purpose fixed screws of fi ing the dimensions of the jet. ne pitch (fig. 16) are convenient. These are set to touch the jet, and then the distance between them can be measured at leisure. The coefficient of velocity is determined directly by measuring the parabolic path of a horizontal jet. Let OX, OY (fig. 17) be horizontal and vertical axes, the origin being at the orifice. Let be the head, and x, y the coordinates of a point A on the parabolic path of the jet. If va is the velocity at j. .1 5 1' 'J

B

FIG. 18. into the waste channel. D is a stilling screen to prevent agitation of the surface at the measuring point, E, and F is a discharge valve for emptying the measuring tank. The rise of level in the tank, the time of the flow and the head over the orifice at that time must be exactly observed. For well made sharp-edged orifices, small relatively to the water surface in the supply reservoir, the coefficients under different conditions of head are pretty exactly known. Suppose the same quantity of water is made to flow in succession through such an orifice and through another orifice of which the coefficient is required, and when the rate of flow is constant the heads over each orifice are noted. Let hi, ha be the heads, wi, Q2 the areas of the orifices, ci, 02 the coefficients. Then since the flow through each orifice is the same Q =61w1'/ (2Zh1) =52w2/ (2§ h2)-52 =C1(wl/402) f (hi/hz) § 19. Coejictents for Bcllmouths and Bellmonthed Orijices.-If an orifice is furnished with a mouthpiece exactly of the form of the a- - -~-~- - -Dfz-zsd —< —o-isD 1.3 D:}1'53'§ @""{ 'offs d 2 -'°~ " t ...... a-aan i Fig. 19. contracted vein, then the whole of the contraction occurs within the mouthpiece, and if the area of the orifice is measured at the smaller end, cf must be put= 1. It is often desirable to bell mouth L the ends of pipes, to avoid the loss of head which occurs if this is not done; and such a bell mouth may also have the form of the con- ' C oejicients of Discharge from Circular Orij'ice r tracted jet. Fig. 19 shows the proportions of such a bell mouth or bell mouthed orifice, which approximates to the form of the contracted jet sufficiently for any practical purpose. the coefficients with different heads.

Temperature 5 1° to 55 °.

iHead in

Diameters of Qrifices in Inches (rl).

h 1 I§ II§ 1I§ i2'2§ |2;|2§ | '§

Head0ver0rifice, inft.=h 1 -66 I-64 11-48 55-77 337-93 Coefficientof velocity=c, , . -959 -967 -975 -994 ~994 ICoefticient0resistance=c, . -087 -069 -052 -012 -012 As there is no contraction after the jet issues from the orifice, c¢= I, c=c, ,; and therefore

Q =ciw/ (21:11) =w~/ {2gh/(1 +c, >}§

20. Coejicients for Sharp-edged or virtually Sharp-edged Orifices.-There are a very large number of measurements of discharge from sharp-edged oritices under different conditions of head. An account of these and a very careful tabulation of the average values of the coefficients will be found in the Hydraulics of the late Hamilton Smith (Wiley & Sons, New York, 1886). The following short table abstracted from a larger one will give a fair notion of how the coefficient varies according to the most trustworthy of the experiments. For such an orifice L. [ lVeisbach found the following values of feet I l

Coefficients (C)

1-0 -613-612 612 611 '6I2 611

-604

-608 -609

-605 -605 -606

-603 -607 -605

-604 -607 -605

-604 I -604 -60

1-50 -610 -612 -6II -606 -610 -607

1-75 -612 -611 -611 -605 -611 -605

-75 -616 -614 -616 '6IO -616 -612 -607'-607 609 2-00 -609 -613 -609 -606 -609 -606 5

I-25 -613 -614 -610 -608'-612 -608

The following table, compiled by ]. T. Fanning (Treatise on Water Supply Engineering), gives values for rectangular orifices in vertical plane surfaces, the head being measured, not immediately over the orifice, where the surface is depressed, but to the still water surface at some distance from the orifice. The values were obtained by graphic interpolation, all the most reliable experiments being plotted and curves drawn so as to average the Coejicient of Discharge for Vertical Circular Orijices, Sharp-edged, with free Discharge into the Air. Q =ca»~/ (zgh). discrepancies.

Coejcients of Discharge for Rectangular Orijices, Sharp-edged, in Vertical Plane Surfaces.

Head Diamefefs of Orifice' éiead tof Ratio of Height to Width. 1

m€:2;L;'; 0o;0 '02 | -04 | -10 I '20 ' -40 1 -60 I I-o 4 2 If 1 % é i A 1 Orifice' Values of C. A A A -5

213 " .657 23; '° ~~ ~~ ~~ reef. 25 5? .5 as -5 55 if “~;£§ 0-6 -655 -630 613 601 -596 588 . “lf 'Sf "li 'Qi 31 of si 0-8 -648 -626 -610 -601 -597 -594 -583 '—I-0 -644 -623 698 699 593 595 '591 0-2 — ~ - ~ - ~ - ~ ~ ~ ~ ~ ~6333 2'0 '632 '614 004 599 599 597 '595 '3 - ~ - - - ~ 6293 '6334 4-0 '623 -699 602 599 593 597 -596 '4 ~ ~ ~ ~ '6140 6306 -6334 8-0 '6I4 605 600 598 597 596 -596 -5 . . . 6050 -6150 6313 -6333 20-0 '60I -599 596 596 596 596 -594 -6 . . -5984 6063 -6156 6317 -6332 -7 . . -5994 6074 -6162. 6319 -6328

At the same time it must be observed that differences of sharpness '8 » '6130 -6999 6032 '6165 6322 '6326 in the edge of the orifice and some other circumstances affect the '9 - - '6134 '6006 6086 '6168 6323 '6324 results, so that the values found by different careful experimenters, 1'0 - - '6135 -6919 '6090 '6172 6320 '6320 are not a little discrepant. When exact measurement of flow has I 1'25 °6188 '6140 -6018 -6095 '6173 6317 '6312 to be made by a sharp-edged orifice it is desirable that the coefficient 1 '50 °6137 '6144 '6026 6100 '6172 6313 '6303 for the particular orifice should be directly determined. 1'75 - '6136 '6145 '6033 6103 '6163 6307 '6296 The following results were obtained by Dr H. T. Bovey in the 2 ~- '6183 '6144 '6036 '6104 '6166 6302 '6291 laboratory of McGill University. Q50

2

Coejicierzt of Discharge for Sharp-edged Orifices. 2-75 -6280 -6173 -6136 -6046 6101 -6155 6274 -6273 3 -6273 -6170 -6132 -6048 6100 -6153 6267 -6267

  • F0¥m 0f 0"f1C@- 3-5 -6250 -6160 '6123 -6050 -6094 -6146 -6254 -6254

S Rectangular Ratio Rectangular Ratioi 4 6245 '6150 '61 10 '6047 6085 '6136 '6236 '6236 quafe- of Sides 4:1. of Sides 161. 4-5 -6226 -6138 -6100 -6044 -6074 -6125 -6222 -6222. He? "1 Ci | ' Long Tm 5 '6208 °6124 -6088 -6038 6063 -61 14 6202 '6202 cular. Sides DIG- 1-0112 Lfmg Long Sides angular. 6 '0158 '6094 '6063 '6020 '6044 '6087 '6154 '0154 C, ,, C, ,| gonal Sides ig? Siqes hor# 7 -6124 -6064 -6038 -601 1 -6032 -6058 -61 IO -61 I4 "°f“fa1- Vefflcal- zO, ml "e"'fa'- zoutal 8 -6090 -6036 -6022 ~6010 -6022 -6033 -6073 -6087 3 » 9 -6060 -6020 '6OI4 -6010 6015 -6020 6045 -6070 1 -6 -6 -6 -6 6 -6 6 -6 6

1 I 62° '627 '628 '642 °643 663 '664 '636 1; -6323 -egig -ei-ii; -egii) cgi; -cgi? 633; -6262 Z 20 -6045 '6024 '6012 -6012 '6014 -6018 6036 -6074 -5 8 -6 8 -6 -6 -6 6 -6 6 -6 8

6 1 607 '614 '616 '626 '627 637 '637 62° -62114 -6234 ~6iii; -cgi; siiis -633; eiiiiii -629; 8 '000 '0I3 '614 '023 '025 034 '035 '019 .35 -6060 -6039 '602I -6014 -6022 -6032 -6049 '6IO3 10 '005 '0I2 '013 '022 '024 032 '033 '0I0 40 -6066 -6045 -6025 -6015 6026 -6037 ~6055 -6114 ii 45 -6054 -6052 -6029 -6016 6030 -6043 -6062 -6125 I6 603 mo 6 62O 622 630 63O 6 7 5o -6086 -6063 -6034 -6018 6035 -6050 -6070 '6140 18 -603 -610 -611 '62O -621 -630 -629 -616

20 -603 -609 »61 1 -620 '62I -629 -628 -616 § 21. Orifices with Edges of Sensible Thickness.-When the edges of The orifice was 0-196 sq. in. area and the reductions were made with g=32-176 the value for Montreal. The value of the coefficient appears to increase as (perimeter) / (area) increases. It decreases as the head increases. It decreases a little as the size of the orifice is greater.

Very careful experiments by I. G. Mair (Proc. Inst. Civ. Eng. lxxxiv.) on the discharge from circular orifices gave the results shown on top of next column.

The edges of the orifices were got up with scrapers to a sharp square edge. The coefficients generally fall as the head increases and as the diameter increases. Professor W. C. Unwin found that the results agree with the formula

c =0-6075+0-0098/4 h-o-0037d,

where h is in feet and d in inches.

the orifice are not bevelled outwards, but have a sensible thickness, the coefficient of discharge is somewhat altered. The following table gives values of the coefficient of discharge for the arrangements of the orifice shown in vertical section at P, Q, R (fig. 20). The plan of all the orifices is shown at S. The planks. forming the orifice and sluice were each 2 in. thick, and the orifices were all 24 in. wide. The heads were measured immediately over the orifice. In this case, Q=cb(H-h)~[{2g(H-{-h)/2}.

§ 22. Partially Suppressed Contract-ion.-Since the contraction of the jet is due to the convergence towards the orifice of the issuing streams, it will be diminished if for any portion of the edge of the orifice the convergence is prevented. Thus, if an internal rim or border is applied to part of the edge of the orifice (fig. 21), the convergence f0r so much of the edge is suppressed. For such cases

G. Bidone found the following empirical formulae applicable:

Table of Coejicients of Discharge for Rectangular Vertical Orifices in Fig. 20. Head 1- i Height of oi-mee, H -/1, in feet. 1 above 1-31 0-66 O'I6 0-10 Orifice ' ' '”“ in feet. P Q R P Q R P Q R P Q R 0-328 0-598 0-644 0-648 0-634 0-665 0-668 0-691 o-664 -666 0-710 0-694 0-696 -656 0 609 0-653 0-657 0-640 O'672 0 675 0-685 0-687 -688 0-696 0-704 0-706 -787 0-612 0-655 0-659 0-641 0-674 0-677 0-684 0-690 692 0-694 0-706 0-708 -984 0 616 0-656 0-660 O'64I 0-675 0-678 0-683 0-693 695 0-692 0-709 0-711 1-968 O°6I8 0-649 0-653 0-640 0-676 0-679, 0-678 0-695 I 697 0-688 0-710 0-712 3-28 0 608 0-632 0-634 0-638 0-674 0-676 0-673, 0-694 -695 0-680 l 0-704 0-705 4'27 0 602 0-624 O'626 0-637 0-673 I 0-675 0-672 0-693 -694 0-678 0-701 0-702 4-92 0-598 | O'62O 0-622 0-637 l 0-673 0-674 0-672 0-692 -693 0-676 0-699 0-699 5-58 0-596 0-618 0-620 0-637 0-672 0-673 0-672 0-692 693 0-676 o-098 0-698 6-56 0-595 0-615 0-617 0-636 0-671 0-672 0-671 0-691 -692 0-675 0-696 0-696 9-84 0-592 0-611 0-612 0-634 0-669 0-670 0-668 0-689 '69O 0-672 O-693 VYYY 0-693 For rectangular oritices Q40-62(1+0-152n and for circular orifxces, c, =0'62): when n is the length of the edge of the orifice over which the border extends, and p is the whole length of edge or perimeter of the orifice. The following are the values of cf, when the border extends over }, i or § of the whole perimeter:- fc C Z/P Rectangular Orifices. Circular cOrifices. 0- 25 0-643 -640 0- 50 0-667 -660 0-75 0-691 -680 For larger values of P

T I TK -ill?: "' ' ':..T.TT '€a F .E .

é u~ f 1 1 | J r ~..; - -~-4 ~=.~~»f¢ - Y .- n/p the formulae a Q 1 M pg lx

Q- .1 5

I L v '.T%§ 'iQ'Q:, ¢ 2 2. - ff/ re not applicable. C. R. Bornemann has shown, however, that these formulae for suppressed contraction are not reliable. § 23. Imperfect Cantraction.-If the sides of the vessel approach near to the edge of the orifice, they interfere with the convergence of the streams to which the contraction is due, and the contraction is then modified. It is generally stated that the influence of the sides begins to be felt if their distance from the edge of 4.1 § 24. Orzjices Furnished with Channels of Discharge.-These external borders to an orifice also modify the contraction. The following coefficients of discharge were obtained with openings 8 in. wide, and small in proportion to the channel of approach (fig. 22, A, B, C). hd" hx in l hi in feet. teen ' - -o656 164 '328 '656 1'640 3'28 4'Q2 6'56 984 A -480 '5II -542 574 -599 -601 -601 -601 -601 B 0-656 -480 -510 -538 506 -592 -600 -602 -602 -601 C, l -527 -553 '574 592 -607 '6IO -610 -609 -608 A I-488 -577 -624 631 -625 -624 -619 -613 -606 B go-164 -487 -571 -606 617 -626 -628 -627 -623 -618 C -585 -614 -633 645 -652 -651 -6504 -650 -649 from a horizontal § 25. Inversion of the Jet.-When a jet issues orifice, or is of small size compared with the head, it presents no l ct, 6

the orifice is less than 2-7 49, ,// times the corresponding /*W A R i i 00%, ,J —f 7- ft ' 4

1 2 . /, ,

I | I V

T I- . .. 1 / /

T ' “ .i 'P l 'l . *= I # ' . 1. v-§ Q | -Q Y4 ' H- ra Q + j; i FIG. 20. FIG. 21. I T FIG- 23- V width of the orifice. The coefficients of contraction for this case | marked peculiarity of form. But if the orifice is in a vertical surare imperfectly known. face, and if its dimensions are not small com ared with the head, P A, B C,

iii
i;; : - -:.;-T-n.;;

"T""" '"'Z'”"" l ffiififflfii Ill hi hi! ', IB i I .1 a 5 = ka 4 I $§ -; -. " i , , ' ' ~—, lil-;§ » -lir;:i;::*"'~ ~~ ' ~ Ji- — rfq '1'53~"1f§ l?57?<§ ', Z ' iI'l". “ "2 Q 'i - ~—10 — — »i l SI/qw 1 in- 20 4 7 '» l i FIG. 22. l<1°""""“'*9"“""""”" J it undergoes a series of singular changes of form after leaving the orifice. These were first investigated by G. Bidone (1781-1839); subsequently H. G. Magnus (1802-1870) measured jets from different orifices; and later Lord Rayleigh (Proc. Roy. Soc. xxix. 71) investigated them anew.

Fig. 23 shows some forms, the upper figure giving the shape of the orifices, and the others sections of the jet. The jet first contracts as described above, in consequence of the convergence of the fluid streams within the vessel, retaining, however, a form similar to that of the orifice. Afterwards it expands into sheets in planes perpendicular to the sides of the orifice. Thus the jet from a triangular orifice expands into three sheets, in planes bisecting at right angles the three sides of the triangle. Generally a jet from an orifice, in the form of a regular polygon of n sides, forms n sheets in planes perpendicular to the sides of the polygon. Bidone explains this by reference to the simpler case of meeting streams. If two equal streams having the same axis, but moving in opposite directions, meet, they spread out into a thin disk normal to the common axis of the streams. If the directions of two streams intersect obliquely they spread into a symmetrical sheet perpendicular to the plane of the streams.

Let al, ag (fig. 24) be two points in an orifice at depths hi, h2 from the free surface. The filaments issuing at ai. az will have the different velocities x/ 2 gh; and / 2gh2.

L - Consequently they will

tend to describe parabolic

paths a, cb1 and azcbz of

  • T

ii:-'l - different horizontal range,

"' /H and intersecting in the

/'s I point 6. But since two

I L filaments cannot simul:”

F Q, taneously flow through the

g Q; 2 same point, they must

exercise mutual pressure,

6, and will be deflected out of

the paths they tend to

1-. describe. It is this mutual

pressure which causes

the expansion of the jet

into sheets.

Lord Rayleigh pointed out that, when the orifices are small and the head is not great, the expansion of the sheets in directions perpendicular to the direction of flow reaches a limit. Sections taken at greater distance from the orifice show a contraction of the sheets until a compact form is reached similar to that at the first contraction. Beyond this point, if the jet retains its coherence, sheets are thrown out again, but in directions bisecting the angles between the previous sheets. Lord Rayleigh accepts an explanation of this contraction first suggested by H. Buff (1805-1878), namely, that it is due to surface tension.

§ 26. Influence of Temperature on Discharge of Oriyices.-Professor W. C. Unwin found (Phil. Mag., October 1878, p. 281) that for sharp-edged orifices temperature has a very small influence on the discharge. For an orifice 1 cm. in diameter with heads of about I to 1% ft. the coefficients were:-

Fro. 24.

Temperature F ....... C.

205° ..... -594

62° ...... -598

For a conoidal or bell-mouthed orifice I cm. diameter the effect of C€ITlp€I'2'tt.ll'C WPIS gl'€3.t€l' i""

Temperature F ....... C.

190; . . 0-987

130° ....... 0-974

60

0' 2

an increase in velocity of discharge of 4% when th24temperature increased 130°.

]. G. iIair repeated these experiments on a much larger scale (Proc. Inst. Civ. Eng. lxxxiv.). For a sharp-edged orifice 2% in. diameter, with a head of I-75 ft., the coefficient was 0-604 at 57° and 0~6o7 at 179° F., a very small difference. With a conoidal orifice the coefficient was 0-961 at 55° and o~981 at 170° F. The corresponding coefficients of resistance are O'O828 and 0-0391, showing that the resistance decreases to about half at the higher temperature.

§ 27. Fire Hose Nozzles.-Experiments have been made by J. R. Freeman on the coefficient of discharge from smooth cone nozzles used for fire purposes. The coefficient was found to be O'983 for %-in. nozzle; o-982 for Q5 in.; o~972 for 1 in.; 0-976 for 1% in.; and o~971 for 1% in. The nozzles were fixed on a taper play-pipe, and the coefficient includes the resistance of this pipe (Amer. Soc. Civ. Eng. xxi.. 1889). Other forms of nozzle were tried such as ring nozzles for which the coefficient was smaller.

IV. THEORY OF THE STEADY MOTION OF FLUIDS. § 28. The general equation of the steady motion of a fluid given under Hydrodynamics furnishes immediately three results as to the distribution of pressure in a stream which may here be assumed. (a) lf the motion is rectilinear and uniform, the variation of Dressure is the same as in a fluid at rest. In a stream flowing in an open channel, for instance, when the effect of eddies produced by the roughness of the sides is neglected, the pressure at each point is simfply the hydrostatic pressure due to the depth below the free sur ace.

(b) If the velocity of the fluid is very small, the distribution of pressure is approximately the same as in a fluid at rest. (c) If the fluid molecules take precisely the accelerations which they would have if independent and submitted only to the external forces, the pressure is uniform. Thus in a jet falling freely in the air the pressure throughout any cross section is uniform and equal to the atmospheric pressure.

(d) In any bounded plane section traversed normally by streams which are rectilinear for a certain distance on either side of the section, the distribution of pressure is the same as in a fluid at rest. DISTRIBUTION or ENERGY IN INco1v11>nEss1B1.a Ftums. § 29. Application of the Principle of the Conservation of Energy to Cases of Stream Line M ation.—The external and internal work done on a mass is equal to the change of kinetic energy produced. In many hydraulic questions this principle is difficult to apply, because from the complicated nature of the motion produced it is difficult to estimate the total kinetic energy generated, and because in some cases the internal work done in overcoming frictional or viscous resistances cannot be ascertained; but in the case of stream line motion it furnishes a simple and important result known as Bernoullis theorem.

Let AB (fig. 25) be any one elementary stream, in a steadily moving fluid mass. Then, from the steadiness of the motion, AB is a fixed path in space through which a stream of fluid is constantly flowing. Let OO be the free surface and XX any horizontal datum line. Let 0 0

A' B B

iff-is, ~ L /-V+

-3{, ..;-..-.- . . ....:. . .; -.2 -.. FIG. 25.

w be the area of a normal cross section, 'v the velocity, p the intensity of pressure, and 2 the elevation above XX, of the elementary stream AB at A, and wi, pi, 111, zi the same quantities at B. Suppose that in a short time t the mass of fluid initially occupying AB comes to A'B'. Then AA', BB' are equal to vt, "vit, and the volumes of fluid AA', BB' are the equal inflow and outfiow=Qt=wvt=w1z:1t, in the given time. If we suppose the filament AB surrounded by other filaments moving with not very different velocities, the frictional or viscous resistance on its surface will be small enough to be neglected, and if the fluid is incompressible no internal work is done in change of volume. Then the work done by external forces will be equal to the kinetic energy produced in the time considered. The normal pressures on the surface of the mass (excluding the ends A, B) are at each point normal to the direction of motion, and do no work. Hence the only external forces to be reckoned are gravity and the pressures on the ends of the stream. The work of gravity when AB falls to A'B' is the same as that of transferring AA' to BB'; that is, GQt (z-21). T he work of the pressures on the ends, reckoning that.at B negative, because it is opposite to the direction of motion, is (pw><'vt) - (p1wiXvit) == Qz(p-pl). The change of kinetic energy in the time t is the difference of the kinetic energy originally possessed by AA' and that finally acquired by BB', for in the intermediate part A'B there is no change of kinetic energy, in consequence of the steadiness of the motion. But the mass of AA' and BB' is GQt/g, and the change of kinetic energy is therefore (GQt/g) (1112/2 -112/2). Equating this to the work done on the mass AB,

GQt(z-zf) -I-Qt(p-pi) = (GQt/g) (viz/2 -112/2). Dividing by GQt and rearranging the terms, . 11"/2g-l~1>/G-|12='vi2/2g+1>i/G4-21; (I) or, as A and B are any two points,

21”/2g+p/G-1-z=constant=H. ' (2)

Now v'/2g is the head due to the velocity v, p/G is the head equivalent to the pressure, and z is the elevation above the datum (see § 16). Hence the terms on the left are the total head due to velocity, pressure, and elevation at a given cross section of the filament, z is easily seen to be the work in foot-pounds which would be done by 1 lb of fluid falling to the datum line, and similarly p/G and vi'/2g are the quantities of work which would be done by I lb of fluid due to the pressure p and velocity 21. The expression on the left of the equation is, therefore, the total energy of the stream at the section considered, per lb of fluid, estimated with reference to the datum line XX. Hence we see that in stream line motion, under the restrictions named above, the total ener y per lb of fluid is uniformly distributed along the stream line. if the free surface of the fluid OO is taken as the datum, and -h, -hi are the depths of A and B measured down from the free surface, the equation takes the form V2/2§ “f'P/G -h =vi2/2g+pi/G -hi; (3) or generally sv'/2 g ~i-p/G - h = constant. (311) § 3o Second Form of the Theorem of Bernoulli Su ose at the pp “ two sections A, B (fig. 262 of an elementary stream small vertical Q pipes are introduced, which may be termed pressure columns; ? f1f q s Bl

'i

~l" ~1:;;: '—~ ”-”—~-~ —i7—5§ 1 'f j, f B' T " if I 6

rf fj Ag *B g

I ' | o ~ - ' it i' ';, f, »f ' -'lf rg C V C 1 El s t FIG. 26. 1£. (§ 8), having their lower ends accurately parallel to the direction of flow. In such tubes the water will rise to heights corresponding to the pressures at A and B. Hence b=p/G, and b'=p-/G. Consequently the tops of the pressure columns A' and B' will be at total heights bl c=p/G+z and b'+c'=p1/G+z1 above the datum line XX. The difference of level of the pressure column tops, or the fall of free surface level between A and B, is therefore E = (St-P 1)/G+(z-Zi); and this by equation (1), 29 is (1112-112)/2g. That is, the fall of free surface level between two sections is equal to the difference of the heights due to the velocities at the sections. The line A'B is sometimes called the line of h draulic gradient, though this term is also used in cases where f>i”iction needs to be taken into account. It is the line the height of which above datum is the sum of the elevation and pressure head at that point, and it falls below a horizontal line A”B" drawn at H ft. above XX by the quantities a =t»2/2g and a' =v,2/2g, when friction is absent. § 31. Illustrations of the Theorem of Bernoulli. In a lecture to the mechanical section of the British Association in 1875, W. Froude gave some experimental illustrations of the principle of Bernoulli. He remarked that it was a common but erroneous impression that a fluid exercises in a contracting pipe A (fig. 27) an excess of pressure against the entire converging surface projected surface as HI, and the pressures parallel to the axis of the pipe, normal to these lgrojected surfaces, balance each other. Similarly the pressures on C, CD balance those on GH, EG. In the same way, in any combination of enlargements and contrac» tions, a balance of pressures, due to the flow of liquid parallel to the 1 F = —-1*—r-, ;—°-""* —" I; I I lr I II I.. v.” 32 €=s I i*.9, §

. -'%»-“

Q ={ 1:2 /// C

  • "|"— 2 E ' / /"'T'E

gf" IN, 5. . ”"r"' 5 -9% f /'§ I " ' Z Z 5' . H 1 - ! 1 Y, /- ! -~1—~ -~~'—}}i})}T?*' "'”

FIG.28.

l axis of the pipe, will be found, provided the sectional area and direction of the ends are the same. The following experiment is interesting. Two cisterns provided with converging pipes were placed so that the jet from one was exactly opposite the entrance to the other. The cisterns being filled I li . 2, . . @ g 1 ' -5 -5- 'f 5 T !::T-: I -e- ~~' S | / " —1- i '/':b. . 15 — -. | / - -~ '3 ", l “Z/, ' I —|— I ' -}> . I G E E Q H L I G I 5; ff ~ ""*"°"'§ T°>——"°%'T>- '— -, — K-.75-> FIG. 29. very nearly to the same level, the 'et from the left-hand cistern A entered the right-hand cistern B (fig. 31), shooting across the free space between them without any waste, except that due to indirectness of aim and want of exact correspondence in the form of the orifices. In the actual experiment there was 18 in. of head in the right and 20% in. of head in the left-hand cistern, so that about which it meets, and that, conversely, A , ' ' A I as it enters an enlargement B, a relief { 3 ' ' ' ";E;' ' ' ' 21"jj2 ' ' ' ' ' ' ' ' ' ' "Pg '°" ' “"'1fI' Q r of pressure is experienced bv the, mg | i -D "“ "'“"L v entire diverrin ~ f f ' ' id -~-'- —- 'J »»»""" g g sur ace o the pipe. | 9 E » Further it is commonly assumed that 9 #aw —- — 5 — .when passing through a contraction A C, there is in the narrow neck an excess of pressure due to the squeezing together of the llquld at that point. These impressions are in no rCSp€Ct Correct; the pressure is smaller as the section of the pipe is smaller and conversely. Fig. 28 shows a pipe so formed that a contraction is followed by an enlargement, and fig. 29 one in which an enlargement is followed by a contraction. The A B Vertical pressure columns

  - show the decrease of

~ pressure at the contraction and increase of pressure at the enlargement. The line abc in both figures shows the C variation of free surface level, supposing the pipe frictionless. In actual pipes, however, work is FIG 27 expended in friction against the pipe; the total head diminishes in proceeding along the pipe, and the free surface level is a line such as abici, falling below abc. Froude further pointed out that, if a pipe contracts and enlarges again to the same size, the resultant pressure on the converging part exactly balances the resultant pressure on the diverging part so that there is no tendency to move the pipe bodily when water flows through it. Thus the conical part AB (fig. 30) presents the same if » 1 FIG. 30. 2% in. were wasted in friction. It will be seen that in the open space between the orifices there was no pressure, except the atmospheric § pressure acting uniformly throughout the system. § 32. Venturi Meter.-An ingenious application of the variation of pressure and velocity in a converging and diverging pipe has been

Fig. 31. made by Clemens Herschel in the construction of what he terms a I Venturi Meter for measuring the flow: in water mains. Suppose that, as in fig. 32, a contraction is made in a water main, the change of section bein radual to avoid the reduction of eddies. The ratio p | g E P of the cross sections at A and B, that is at inlet and throat, is in actual meters 5 to I to 20 to I, and is very carefully determined by the maker of the meter. Then, if 11 and u are the velocities at A and B, -u=pv. Let pressure pipes be introduced at A, B and C,

FIG. 32. and let H1, H, H2 be the pressure heads at those points. Since the velocity at B is greater than at A the pressure will be less. Neglecting friction H1-H12/2g = H -l-U2/2g, H1-H = (ui-02)/2g = (p2-1)v2/2g. Let h = H, -H be termed the Venturi head, then 11 =~/ {p'.2gh/(p“-1)}, from which the velocity through the throat and the discharge of the main can be calculated if the areas at A and B are known and h observed. Thus if the diameters at A and B are 4 and 12 in., the areas are I2'57 and II3°I sq. in., and p=9, u=/ 81/804 (zgh) = I°O07/ (zgh). If the observed Venturi head is 12 ft., u=28 ft. per sec., and the discharge of the main is 28><12~57=351 cub. ft. per sec. Hence by a simple observation of pressure difference, the How in the main-at any moment can be determined. Notice that the pressure height at C will be the same as at A except for a small loss h/ due to friction and eddying between A and B. To get the pressure at the throat very exactly Herschel surrounds it by an annular passage communicating with the throat by several small holes, sometimes formed in vulcanise to prevent corrosion. Though constructed to prevent eddying as much as possiole there is some eddy loss The main effect of this is to cause a loss of head between A and C which may vary from a fraction of a foot to perhaps 5 ft. at the highest velocities at which a meter can be used. The eddying also affects a little the Venturi head h. Consequently an experimental coefficient must be determined for each meter by tank measurement. The range of this coefficient is, however, surprisingly small. If to allow for friction, u=k~/lp2/(pi-1)}/(zgh), then Herschel found values of Wk from o~97 to I~0 for throat velocities varying from 8 to 28 ft. per sec. The meter is extremely convenient. At Staines reservoirs there are two meters of this type on mains 94 in. | in diameter. Herschel contrived a recording arrangement which records the variation of flow from hour to hour and also the total flow in any given time. In Great Britain 'ie meter is constructed by 3. Kent, who has made improvements in the recording arrangement. In the Deacon Waste 'Vater Meter (Hg. 33) a different principle is used. A disk D, partly counterweight, is water flowmain in a The un- of the disk the im 'ict 4..~ W /If ll @ 'H/as A I ¢ ? /filfi Outlet; 1-G( 0,01 J

/ //ll x( k ¥ § 0 balanced by a suspended in the ing through the conical chamber. balanced weight is supported by pt of the water. If the discharge of the main increases the disk rises, but as it rises its position in the chamber is such that in consequence of the larger area the velocity is less. It finds, therefore, a new position of equilibrium. A pencil P records on a drum moved by clockwork Ehe position of the disk, and from this the variation of flow is inerre § 33. Pressure, Velocity and Energy in Dqferenl Stream Lineng P ¢ E

-, ,,
 I

ws ;..., 'CL. 3 2 ¢, , -. »; . , . . § sl . I %|a Fxo. 33. The equation of Bernoulli gives the variation of pressure and velocity from point to point along a stream line, and shows that the total energy of the flow across any two sections is the same. Two other directions may be defined, one normal to the stream line and in the plane containing its radius of curvature at any point, the other normal to the stream line and the radius of curvature. For the problems most practically useful it will be sufficient to consider the stream lines as parallel to a vertical or horizontal plane. If the motion is in a vertical plane, the action of gravity must be taken into the reckoning; if the motion is in a horizontal plane, the terms expressing variation of elevation of the filament will disappear Let AB, CD (fig. 34) be two consecutive stream lines, at present assumed to be in a vertical plane, and PQ a normal to these lines 1>+'1P ¢ I I I v+dv C' f- / Q 7->v P I D


5 | ' r 1 f'~¢>~*1 1 I 5 of FIG. 34. making an angle ¢ with the vertical. Let P, Q be two particles moving along these lines at a distance PQ=ds, and let z be the height of Q above the horizontal plane with reference to which the energy is measured, 11 its velocity, and p its pressure. Then, if H is the total energy at Q per unit of weight of Huirl, H=z+p/G-H12/2g. Dififerentiating, we get dH =dz-1-dp/G-1-vdv/g, (1) for the increment of energy between Q and P. But dz=PQ cos q§ =ds cos ¢>; dH =dp/G-Q-vdv/g+ds cos ¢>, (Ia) where the last term disappears if the motion is in a horizontal plane. Now imagine a small cylinder of section w described round PQ as an axis. This will be in equilibrium under the action of its centrifugal force, its weight and the pressure on its ends. But its volume is ods and its weight Gods. Hence, taking the components of the forces parallel to PQwdp = Gvzwds/gp-Gw cos ¢>ds, where p is the radius of curvature of the stream line at Q. Consequently, introducing these values in (I), dH = v2ds/gp +°vdv/g = (v/g) (v/p -}-dv/ds)ds. (2) CURRENTS § 34. Rectilinear Current.-Suppose the motion is in parallel straight stream lilies (fig. 35) in a vertical plane. Then p is infinite, and from eq. (2), § 33, dH évdz//g. Comparing this with (I) we see that dz--dp/G=o; I z-+-p/G =constant; (3) or the pressure varies hydro statically as in a fluid at rest. For two stream lines in a horizontal plane, z is constant, and there- Y fore p is constant. A Ti Radiating Current.-Suppose E P I u I 1 “E” § water flowing radially between, I E horizontal parallel planes, at 'lf Q . l 3 a distance apart=5. Conceive two cylindrical sections of the current at radii rl and M, where the velocities are '11, and 112, and flow across each cylindrical section o the current is the same, Q =27T'1'15U1 =27f'1'257}2 V1771 = hi);- fi/V2 = 112/1/1. (4) The following theorem is taken from a paper by ]. H. Cotterill, “ On the Distribution of Energy in a Mass of Fluid in Steady Motion, " Phil. Mag., February 1876. FIG.35. the pressures pl and pg. Since the I . 1 The velocity would be infinite at radius 0, if the current could be conceived to extend to the axis. Now, if the motion is steady, H = pi/G -I-vf/2g = P2/G +v2”/2g; = P2/G 'i”7i2Z'12/"222g; (P2'°P1l/Q =i'i2<I'f1j2/ffl/3§ = l'l'7'12L'1:/}'2Z2g. Hence the pressure increases from the interior outwards, in a way indicated by the pressure columns in fig, 36, the curve through the free surfaces of the pressure columns being, in a radial section, the quasi-hyperbola of the form xy"=c3. This curve is asymptotic to a I horizontal line, H ft. above the line from which the pressures are measured, and to the axis of the current. Free Circular Vortex.-A free circular vortex is a revolving mass of water, in which the stream lines are concentric circles, and in which

Fig. 36. the total head for each stream line is the same. Hence, if by any slow radial motion portions of the water strayed from one stream line to another, they would take freely the velocities proper to their new positions under the action of the existing fluid pressures only. For such a current, the motion being horizontal, we have for all the circular elementary streams f H =p/G -I-U2/2g =constant; dH = dp/G -I-adv/g = 0. (7) Consider two stream lines at radii r and r-I~dr (fig. 36). Then in (2), § 33, p=r and <ls=dr, wfdr/gr-I-vdv/g=o, do/1' = -dr/r, 1; no I /r, (8) precisely as in a radiating current; and hence the distribution of pressure is the same, and formulae 5 and 6 are applicable to this case. Free Spiral Vorlex.-As in a radiating and circular current the equations of motion are the same, they will also apply to a vortex in which the motion is compounded of these motions in any proportions, provided the radial component of the motion varies inversely as the radius as in a radial current, and the tangential component varies inversely as the radius as in a free vortex. Then the whole velocity at any point will be inversely proportional to the radius of the point, and the fluid will describe stream lines having a constant inclination to the radius drawn to the axis of tl, e current. That is, the stream lines will be logarithmic spirals. When water is delivered from the circumference of a centrifugal iipmp or turbine into a chamber, it forms a free vortex of this kind. he water flows spirally outwards, its velocity diminishing and its 4-5 pressure increasing according to the law stated above, and the head along each spiral stream line is constant. § 35. Forced Vortexr-If the law of motion in a rotating current is different from that in a free vortex, some force must be applied to cause the variation of velocity. The simplest case is that of a rotating current in which all the particles have equal angular velocity, as for instance when they are driven round by radiating paddles revolving uniformly. Then in equation (2), § 33, considering two circular stream lines of radii r and r-l-dr (ng. 37), we have p=r, ds= dr. If the angular velocity is ai, then 1J= ar and do =adr. Hence dH = afrdr/g-I~o.2rdr/g =2a2rdr/g. Comparing this with (1), § 33, and putting dz=-o, because the motion is horizontal, dp/G+a'lr¢lr/g=2a2rdr/g, dp/G = afrdr/g, (> 11/G = ag#/2g-I-constant. 9 Let Pi, ri, vi be the pressure, radius and velocity of one cylindrical SCCUOI1, Pz, r2, U2 those of anotlier; then 91/Q-agrf/2g == /12/(Q-0.2132/Zg; <P2 p1)/C' = f1“(f'22'7'12l/ZS = (i'22'@'12)/2£- (10) pressure increases from within outwards in a curve I That is-, the ax I I

I I

I / I | / | I / I // I ~ c 1 / I ' s / 1 1 5 § l / fi E 1—1-l / e f =~g-ff 1; f QI il if- I r F I I 1 1 I /, jj;; /// // //, ' d - // | ;/'Q 1 // » # 1 I /11" J;*, -~' Ll I li ' Il

% /'/

/ /// 3 , }” ///

//

W 1, '/ ', / I I I F1<;.37. which in radial sections is a parabola, and surfaces of equal pressure are paraboloids of revolution (rig. 37). D1ssiPAT1o:~1 or HEAD IN SHOCK § 36. Relation of Pressure and Velocity in a Stream in Steady llloliorz when the Changes of Section of the Slream are Abrupt./Vhen a stream changes section abruptly, rotating eddies are formed which dissipate energy. The energy absorbed in producing rotation is at once abstracted from that effective in causing the flow, and sooner or later it is Wasted by frictional resistances due to the rapid relative motion of the eddying parts of the fluid. In such cases the work thus expended internally in the fluid is too important to be neglected, and the energy thus lost is commonly termed energy lost in shock. Suppose fig. 38 to re resent a stream having such an abrupt change of section. Let Ali CD be normal sections at points where ordinary stream line motion has not been disturbed and where it has been re-established. Let rn, p, v be the area of section. pressure and velocity at AB, and wl, pl, vi corresponding quantities at CD. Then if no work were expended internally, and assuming the stream horizontal, we should have P/G +112/2g = 1>i/G+v1'/2g- (I) This differs from the expression (I), § 290btained for cases where

But if work is expended in producing irregular eddying motion, the head at the section CD will be diminished. Suppose the mass ABCD comes in a short time Z to A'B'C'D'. The resultant force parallel to the axis of the stream is Pw'l'Po(wi-0->)*P1w1» where po is put for the unknown pressure on the annular space between AB and EF. The impulse of that force is lPw+Po(<°1“w>'P1'»°1l3l'he horizontal change of momentum in the same time is the difference of the momenta of E C '" CDC'D' and ABA'B', because the amount / of momentum be-A / tween A'B' and CD J;//' 5: remains unchanged if the motion is 1-T' -' '! i steady. The volume — — #4 4 ' of ABA'B'orCDC'D', B ..f beiiglg the infiqow and

out ow in the time
  l, is Qt=0J'Ul=0J1'U1t,

5- ' and the momentum of DD' these masses is FIG 33 (G/g)Q1;tand(G/g)Q'z/lt. The change of momentum is therefore (G/g)Qt(vi-v). Equating this to the impulse, lP°'+P0('”1'°°)'P1°~'1l¢ = (G/§)Qf§ 7»'1"»')-Assume that po =p, the pressure at AB extending unchanged through the portions of fluid in contact with AE BF which lie out of path of the stream. Then (since Q=w11/1) i t e (P-Pi) = (G/g)vi(1'1-ff) 3 P/G-P1/G =°v1(v1~v)/g; (2) P/G +v'/2g =1>1/G +v12/2g+ (v-v»)“/2g- (3) no sensible internal work is done, by the last term on the right. That is, (7)-7J1)2/2g has to be added to the total head at CD, which is p, /G +2112/2g, to make it equal to the total head at AB, or (v-v,)'=/2g is the head lost in shock at the abrupt change of section. But 1'-rl is the relative velocity of the two parts of the stream. Hence, when an abrupt change of section occurs, the head due to the relative velocity is lost in shock, or (v-'v1)'/2g foot-pounds of energy is wasted for each pound of fluid. Experiment verifies this result, so that the assumption that po=p appears to be admissible, If there is no shock, P1/G = P/G 'l' (1/2"1/12)/2% P1/'G:P/G-Ui('U1-11)/§ Hence the pressure head at CD in the second case is less than in the former by the quantity (v-'v1)2/2g, or, putting wit/1 =wv, by the quantity lf there is shock, (v”/2g)(I-<»/w1)”- (4) ', THEORY OF THE DISCHARGE FROM ORIFICES AND MOUTHPIECES § 37. .Minimum Coejicient of Contraction. Re-entrant Mouth-Suppose that in a short time t, the mass 00110 comes to the position ()'O” a'a'; the impulse of the horizontal external forces acting on the mass during that time is equal to the horizontal change of momentum. The pressure on the side OC of the mass will be balanced by the pressure on the opposite side OE, and so for all other portions of the vertical surfaces of the mass, excepting the portion EF opposite the mouthpiece and the surface AaaB of the jet. On EF the pressure is simply the hydrostatic pressure due to the depth, that is, (pa-i-Gh)S'l. On the surface and section Atw, B of the jet, the horizontal resultant of the pressure is equal to the atmospheric pressure pa acting on the vertical projection AB of the jet; that is, the resultant pressure is -Pail. Hence the resultant horizontal force for the whole mass OOaa is (pa+Gh)S2-paSZ=GhSZ. lts impulse in the time tis Ghfl Z. Since the motion is steady there is no change of momentum between O'O' and aa. The change of horizontal momentum is, therefore, the difference of the horizontal momentum lost in the space OOO'O and gained in the space aaa'a'. In the former space there is no horizontal momentum. The volume of the space aaa'a' is owl; the mass of liquid in that space is (G/g)w'vt; its momentum is (G/g)wv2t. Equating impulse to momentum gained, Ghilt = (G / g)w'v2t; w/SZ =gh/112. But v2=2gh, and w/§ Z=r., ; w/Q=§ =C¢; a. result confirmed by experiment with mouthpieces of this kind. A similar theoretical investigation is not possible for orifices in plane surfaces, because the velocity along the sides of the vessel in the neighbourhood of the orifice is not so small that it can be neglected. The resultant horizontal pressure is therefore greater than Ghfl, and the contraction is less. The experimental values of the coefficient of discharge for a re-entrant mouthpiece are o~5149 (Borda), o-5547 (Bidone), 0-5324 (Weisbach), values which differ some point in the interior of the vessel from which the fluid is discharged, and gradually acquire the velocity of the jet. Let Mm, fig. 40 be such a filament, the point M being taken where the Velocity is insensibly small, and m at the most contracted section of the jet, where the filaments have become parallel and 9~ ET

mem

E w"7 D-9°é'” § ”s5J 9¢'“>9, 0 §§ °1

s§ <

5:3 >-1$C ~<:-fn ts? ru '<-I mi B.s."3. £f=x< C7 gem =>' sv E'§ <r

-wg

als § = > |'fv ¢. f'J an F9'>¢. OID O § <>

B

QT! go. B Ii

""°"' """”i . ..... . 7i Y . 1 M K &'§ ¥§ @ wr t H. It 4 L rn, h »<mzwz ///W/z/ww F IG. 40.

1 —.. .Y . l F ——

, v

B D can be determined theoretically and as it is the case where the convergence of the streams approaching the orifice takes place through the greatest possible angle the co efficient thus deter mined is the minimum coefficient Let fig 39 represent a vessel with vertical sides OO being the free water surface at which the pressure is pa. Suppose the liquid issues by a horizontal mouthpiece, which is re-entrant and of the greatest length which permits the jet to spring clear from the inner end of the orifice, without adherpzwfv of Bordar-ln one special case the coefficient of contraction r- A ' D . I ., ~ a'-"1 -"—0' . 1 1 1 . |, . I l . . . I . | ~

h ' .

y . r i Y ¢ P A C E i -L Q l-~ - - AA-f* -»- -Y-FIG. 39. ing to its sides. With such an orifice the velocity near the pointsCD is negligible, and the pressure at those points may be taken equal to the hydrostatic pressure due to the depth from the free surface. Let SZ be the area of the mouthpiece AB, w that of the contracted jet aa exercise uniform mutual pressure. Take the free surface AB for datum line, and let pl, zu, hi, be the pressure, velocity and depth below datum at M; p, 11, h, the corresponding quantities at m, Then § 29, eq. (311), 11,2/2g-hp, /G~h, = v2/2g-ijp/G-k. (I) But at M, since the velocity is insensible, the pressure is the hydrostatic pressure due to the depth; that is, 'v1=o, p1=p, ,-l-Ghl. At m, 1>=pa, the atmospheric pressure round the jet. Hence, inserting these values, 0+I>a/GTh1'h1=U2/2§ +Pf1/C415 v*/2g=h; (2) or v=/ (zgh) =8-o25/ h. (za) That is, neglecting the viscosity of the fluid, the velocity of filaments at the contracted section of the jet is simply the velocity due to the difference of level /4 of the free surface in the, reservoir and the orifice. if - — If the orifice is small in 3 5'; dimensions compared with 1 M I h, the filaments will all l I have nearly the same vel- l ], I ., ocity, and if h is measured l- a 3 to the centre of the orifice, 1 ' 1 w the equation above gives , """ b""" the mean velocity of the l I5 jet 1 .. . a | .-Case of a Submerged 1 Orifice.-Let the orifice 'E "ii "W discharge below the level FIG. 41. of the tail water. Then using the notation shown in fig. 41, we have at l'I, vi =o, p1 =Gh;-l-pf. at m, p=Gh;, +pa. Inserting these values in (3), § 29, 0-{-hi-l-pa/G-h, =v2/2g-l-h3-hg-l-11a/G; v2/2g=h2—h3=h. (3) where h is the difference of level of the head and tail water, and may be termed the effective head producing flow.

Hydraulics Fig. 42.JPG
Fig. 42.

Case where the Pressures are different on the Free Surface and at the Orifice.—Let the fluid flow from a vessel in which the pressure is p0 into a vessel in which the pressure is p, fig. 42. The pressure p0 will produce the same effect as a layer of fluid of thickness p0/G added to the head water; and the pressure p, will produce the same effect as a layer of thickness p/G added to the tail water. Hence the effective difference of level, or effective head producing flow, will be

h = h0 + p0/G − p/G;

and the velocity of discharge will be

v = √ [ 2g { h0 + (p0 − p) / G } ].
(4)

We may express this result by saying that differences of pressure at the free surface and at the orifice are to be reckoned as part of the effective head.

Hence in all cases thus far treated the velocity of the jet is the velocity due to the effective head, and the discharge, allowing for contraction of the jet, is

Q = cωv = cω √ (2gh),
(5)

where ω is the area of the orifice, cω the area of the contracted section of the jet, and h the effective head measured to the centre of the orifice. If h and ω are taken in feet, Q is in cubic feet per second.

It is obvious, however, that this formula assumes that all the filaments have sensibly the same velocity. That will be true for horizontal orifices, and very approximately true in other cases, if the dimensions of the orifice are not large compared with the head h. In large orifices in say a vertical surface, the value of h is different for different filaments, and then the velocity of different filaments is not sensibly the same.

Simple Orifices—Head Constant

Hydraulics Fig. 43.JPG
Fig. 43.

§ 39. Large Rectangular Jets from Orifices in Vertical Plane Surfaces.—Let an orifice in a vertical plane surface be so formed that it produces a jet having a rectangular contracted section with vertical and horizontal sides. Let b (fig. 43) be the breadth of the jet, h1 and h2 the depths below the free surface of its upper and lower surfaces. Consider a lamina of the jet between the depths h and h + dh. Its normal section is bdh, and the velocity of discharge √2gh. The discharge per second in this lamina is therefore b√2gh dh, and that of the whole jet is therefore

Q = h2h1 b √ (2gh) dh
= 2/3 b √2g { h23/2 − h13/2 },
(6)

where the first factor on the right is a coefficient depending on the form of the orifice.

Now an orifice producing a rectangular jet must itself be very approximately rectangular. Let B be the breadth, H1, H2, the depths to the upper and lower edges of the orifice. Put

b (h23/2 − h13/2) / B (H23/2 − H13/2) = c.
(7)

Then the discharge, in terms of the dimensions of the orifice, instead of those of the jet, is

Q = 2/3 cB √2g (H23/2 − H13/2),
(8)

the formula commonly given for the discharge of rectangular orifices. The coefficient c is not, however, simply the coefficient of contraction, the value of which is

b (h2 − h1) / B (H2 − H1),

and not that given in (7). It cannot be assumed, therefore, that c in equation (8) is constant, and in fact it is found to vary for different values of B/H2 and B/H1, and must be ascertained experimentally.

Relation between the Expressions (5) and (8).—For a rectangular orifice the area of the orifice is ω = B(H2 − H1), and the depth measured to its centre is ½ (H2 + H1). Putting these values in (5),

Q1 = cB (H2 − H1) √ {g (H2 + H1) }.

From (8) the discharge is

Q2 = 2/3 cB √2g (H23/2 − H13/2).

Hence, for the same value of c in the two cases,

Q2/Q1 = 2/3 (H23/2 − H13/2) / [ (H2 − H1) √ { (H2 + H1)/2} ].

Let H1/H2 = σ, then

Q2/Q1 = 0.9427 (1 − σ3/2) / {1 − σ √ (1 + σ) }.
(9)

If H1 varies from 0 to ∞, σ( = H1/H2) varies from 0 to 1. The following table gives values of the two estimates of the discharge for different values of σ:—

H1/H2 = σ. Q2/Q1. H1/H2 = σ. Q2/Q1.
0.0 .943 0.8 .999
0.2 .979 0.9 .999
0.5 .995 1.0 1.000
0.7 .998    

Hence it is obvious that, except for very small values of σ, the simpler equation (5) gives values sensibly identical with those of (8). When σ < 0.5 it is better to use equation (8) with values of c determined experimentally for the particular proportions of orifice which are in question.

Hydraulics Fig. 44.JPG
Fig. 44.

§ 40. Large Jets having a Circular Section from Orifices in a Vertical Plane Surface.—Let fig. 44 represent the section of the jet, OO being the free surface level in the reservoir. The discharge through the horizontal strip aabb, of breadth aa = b, between the depths h1 + y and h1 + y + dy, is

dQ = b √ {2g (h1 + y) } dy.

The whole discharge of the jet is

Q = d0 b √ { 2g (h1 + y) } dy.

But b = d sin φ; y = ½d (1 − cos φ); dy = ½d sin φ dφ. Let ε = d/(2h1 + d), then

Q = ½d2 √ { 2g (h1 + d/2) } π0 sin2 φ1 − ε cos φ dφ.

From eq. (5), putting ω = πd2/4, h = h1 + d/2, c = 1 when d is the diameter of the jet and not that of the orifice,

Q1 = 1/4πd2 √ {2g (h1 + d/2) },
Q/Q1 = 2/π π0 sin2 φ √ {1 − ε cos φ} dφ.

For

h1 = ∞, ε = 0 and Q/Q1 = 1;

and for

h1 = 0, ε = 1 and Q/Q1 = 0.96.

So that in this case also the difference between the simple formula (5) and the formula above, in which the variation of head at different parts of the orifice is taken into account, is very small.

Notches and Weirs

§ 41. Notches, Weirs and Byewashes.—A notch is an orifice extending up to the free surface level in the reservoir from which the discharge takes place. A weir is a structure over which the water flows, the discharge being in the same conditions as for a notch. The formula of discharge for an orifice of this kind is ordinarily deduced by putting H1 = 0 in the formula for the corresponding orifice, obtained as in the preceding section. Thus for a rectangular notch, put H1 = 0 in (8). Then

Q = 2/3 cB √(2g) H3/2,
(11)

where H is put for the depth to the crest of the weir or the bottom of the notch. Fig. 45 shows the mode in which the discharge occurs in the case of a rectangular notch or weir with a level crest. As, the free surface level falls very sensibly near the notch, the head H should be measured at some distance back from the notch, at a point where the velocity of the water is very small.

Since the area of the notch opening is BH, the above formula is of the form

Q = c × BH × k √(2gH),

where k is a factor depending on the form of the notch and expressing the ratio of the mean velocity of discharge to the velocity due to the depth H.

Fig. 45.

§ 42. Francis’s Formula for Rectangular Notches.—The jet discharged through a rectangular notch has a section smaller than BH,

(a) because of the fall of the water surface from the point where H

FIG. 45

is measured towards the weir, (b) in consequence of the crest con- l or, introducing a coefficient to allow for contraction, traction, (0) in consequence of the end contractions. It may be l Q=Cl5/ (2g)Hi, pointed out that while the diminution of the section of the jet due When a notch is used to gauge a stream of varying flow, the ratio to the surface fall and

B/H varies if the notch is rectangular, but is constant if the notch is to the crest contraetiltln triaéiigular. £I'his led Professor James Thomson to suspect that the “" **'*§ ";§ "t:=»*f is proportion to the coe cient o disl" length of the weir the charge c would

A * m ', ,', ., ,, , ' /71 end contractions have be mucli more lé B ii-S li&;lra*=, l ~ll'H, ~, ;f.'§ i, 'lf', ;{/, '/.1-, lf Age? nearly the same effect, constant with;§ , § 3f;§ f;:;r f lg .ll'1'jl-llf:{l1;~{ ll., l, 'llf!11ll, l"l' l, ff whether the we1r is wlde different values 1; 5" 1

", ' ' gf” or narrow. of H in a trian- "11' '.'.'.Z"' "" f

% '7, ,»'{ 4; », f § '*l J. B. Franc1s'sexper1- gular than in a H % @1')'fh~§ |ll ll, Q 1.11-if ments showed that a r e c t a n g u l a r 31; ' ., ; }ifi, l'H;4lV~, l' l'., l |'| perfect end contraction, notch, and this 0 I

.'f;.!' '{, § fi1|, f, ?ll, ll l== - Hg when the heads varied ' has been experi- .L

5 " ' i" l i § "l'liiIl from 3 to 24 in., and mentally shown """' "'""-M

, ,.€'§ a=¢
;;E11e.-sei; Q% U16 1@11§ th Of the W@if to be the Case- FIG 46,
  1. Fig was not less than three Hence a triantingesd

thj headé diinin- gular 1no%h is mire suit lab lie for accurate gaugirégs Fhan ajrecltingular WV, % X 15 6 10 6 €Q '1V€ notc or a s arp-e fe triangu ar notcn ro essor . omson / 4 M length of the weir by ' found t=o-617. It wilf be seen, as in § 41, that since § BH is the an 211110L111t aPp1'OX1- area of section of the stream through the notch, the formula is ¢ mTm t 'L Wmvwwlww mately equal to one- ' again of the form 1 T' 4 1 3.122 -“Bids l'i;£'.; Q="<fBH><k“<2@H>" i=' '/=T "*** l 5 ' Cl =f§ s ' h ' f he ea vel it in the n teh to the //§ N, . ' where k, is the ratio o t m n oc y 0 —* —~- - —— 53? W (Ig thi? nolgchéw Weir' ang 4 velocity atithe depth H. It may easily be shown that for all notches QW/ b hi 3 Ca n?eaST1re the discharge can be expressed in this form. M '“ fi 1” th? Wcuil W 'Fifi § 44. Weir with ct Broad Sloping Crest.-Suppose a weir formed |l ll the 'Va-LCV lfgeir yfstlh ' with a broad crest so sloped that the streams flowing over it have a ll, ll', w i SH t? “ht Oh t, 9, movement sensibly rectilinear and uniform (Fig. 47). Let the inner M f§ Jet Pssslnlidtbrfiug tile edge be so rounded as to prevent a crest contraction. Consider a Q ~ ;§ ;“)ii§ ;'“fO1 etv;)0';1(i filament aa', the point a being so far back from the weir that the /+f'.:"'» ki ' ' 1

3:5-: ':1»»'1;~E='é. l, & contractions. In a weir 3, ' ' ~ - . 0

ra T., — *as ain, /%4. 'j, ; <1 1

"' ' "" ' '/ ' /f " l' end contractions. ff gs?- Hence, generally, the

width of the jet is l-o- InH, where n is the number of end contractions of the stream. The contractions due to the fall of surface and to the crest contraction are proportional to the width of the jet. Hence, if CH is the thickness of the stream over the Weir, measured at the contracted section, the section of the jet will be c(l-o-InH)H and (§ 41) the mean velocity will be § /(2gH). Consequently the discharge will be given by an equation of the form

Q=§ c(l-o-InH)H/2gH

=5-g,5a(l-o-1nH, H§ .

This is Francis's formula, in which the coefficient of discharge c is much more nearly constant for different values of l and h than in the ordinary formula. Francis found for c the mean value O~622, the weir being sharp-edged.

§ 43. Triangular Notch (fig. 46).-Consider a lamina issuing between the depths h and h-l-dh. Its area, neglecting contraction, will be bdh, and the velocity at that depth is xl (2gh). Hence the discharge for this lamina is

bx/E1 dh.

But B/b=H/(H-h); b=B(H-h)/H.

Hence discharge of lamina

= B(H -h) / (2gh)dh/H;

and total discharge of notch H

=Q = B~/(2.11) o (H—h)hédh/H

=f>'5B/ (2g)H§ .

iiii fiii

/ / Q "“S&.¢;g=, ..b"""

Ww%%WW, %rAl/r%o;, , t;, nMW/ //r/M

Fig. 47.

velocity of approach is negligible. Let OO be the surface level in the reservoir, and let fl be at a height /i” below OO, and h' above a'. Let h be the distance from OO to the weir crest and e the thickness of the stream upon it. Neglecting atmospheric pressure, which has no influence, the pressure at (L is Gh”; at a' it is Gz. If v be the velocity at of,

v2/2g=h'+h”-z=h-e;

Q =be /2g(h-e).

Theory does not furnish a value for e, but Q==0 for e=o and for e =h. Q has therefore a maximum for a value of e between o and h, obtained by equating dQ/de to zero. This gives e = § h, and, inserting this value,

Q =o-385 bh / zgh,

as a maximum value of the discharge with the conditions assigned. Experiment shows that the actual discharge is very approximately equal to this maximum, and the formula is more legitimately applicable to the discharge over broad-crested weirs and to cases such as the discharge with free upper surface through large masonry Coefficients for the Discharge over VI/eirs, derived from the Evperirnents of T. E. Blackwell. VI/hen rnore than one experiment was made with the fame head, and the results were pretty uniform, the resulting coejlcients are marked with an (*). The eject of the converging wing-boards is very strongly marked.

li Headi in Sharp Edgei Planks 2 in. thick, square on Crest. Crests 3 ft. wide. inchcsd 4 it I

iiltiriiilgihl miiiig-bdiiiiis 3 ft, long, 3 ft. long, 3 ft. long, 6 ft. long, ro ft.long, ro ft. long Water in 3 it' long' mit- long' 3 ft- iong- 6 ft' long' 1°h"l"ng' making an angle level. fall 1 in 18. fall 1 in 12. level. level. fall I in x8 Reservoir. 0 6o°.

I '677 309 '457 '459 '435 ' '754 452 545 467 - ~ 331 467 2 °675 803 -5119* '561 5155* -675 482 546 533 - - 479* '495* 3 '630 642* -563* -597* 569* ~ - 441 537 539 '492* . . . . 4 '517 656 '549 '575 602* '656 419 431 455 '497* -~ ~ 515 5 -602 650* -588 -6or* 6o9* -671 479 ~516 .. . . 518 6, -593 — -sas* ~6<>8* 576* -~ 501* ~~ 531 ~s07 513 543 Z; -608: 57§ ;' .. ~488 ~513 -527 -497 .. - - 5 I ' 0 '590 54 '470 '491 - » ~ - 4 50

9 530 -600 -569* ~558* -476 -492* '498 -480* 486 . . I0 - '614* '539 -5s4* -~ »- -~ 4651 455 ..

12 .. .., . 'g25* -534* . . . -467 ..

I4 .., . -49 . g . .. . .. .. .., .,

The discharge per second varied from -461 to -665 cub. ft in two experiments. The coefficient '435 is derived from the mean value sluice openings than the ordinary weir formula for sharp-edged weirs. It should be remembered, however, that the friction on the sides and crest of the weir has been neglected, and that this tends to reduce a little the discharge. The formula is equivalent to the ordinary weir formula with e =o-577. SPECIAL Cases or DISCHARGE FROM ORIFICES § 45. Cases in which the Velocity of Approach needs to be taken into Account. Rectangular Oriflces and Notches:—In finding the velocity at the orifice in the preceding investigations, it has been assumed that the head h has been measured from the free surface of still water above the orifice. In many cases which occur in practice the channel of approach to an orifice or notch is not so large, relatively to the stream through the orifice or notch, that the velocity in it can be disregarded. Let hi, ho (fig. 48) be the heads measured from the free surface to the top and bottom edges of a rectangular orifice, at a point in the E?E;Zq?¥sE, ........... . .. ........., .... 4 " “;i;ii: F2

T-I'4'-3'- -' '“?§ ?f'F="-; i, ':,

7 Ti if; -|

2 41

w he

))»; ... .> . I

I

I

|

-... lf, . .

FIG. 48.,

channel of approach where the velocity is u. It is obvious that a fall of the free surface,

f>= M”/2g

has been somewhere expended in producing the velocity u, and hence the true heads measured in still water would have been lz1+f) and h2+b. Consequently the discharge, allowing for the velocity of approach, is

Q=§ ¢b~/2§ l(hfz+f))*-(711+fJ>3l- (I) And for a rectangular notch Q- which hi=0, the discharge is Q=§ ¢b/ 2§ {(ll2 +f))i-f>3}~ (2) In cases where u can be directly determined, these formulae give the discharge quite simply. Wl1en, however, u is only known as a function of the section of the stream in the channel of approach, they become complicated. Let SZ be the sectional area of the channel where hx and la are measured. Then u=Q/ S2 and f)=Q2/2g $22. This value introduced in the equations above would render them excessively cumbrous. In cases therefore where SZ only is known, it is best to proceed by approximation. Calculate an approximate value Q' of Q by the equation

Q' = § cb/ Zglhgg -h13}.

Then b= Q'2/2gQ” nearly. This value of Y) introduced in the equations above will give a second and much more approximate value of § 46. Partially Submerged Rectangular Orifices and Notchesf-When the tail water is above the lower but below the upper edge of the orifice, the flow in the two parts of the orifice, into which it is divided by the surface of the tail water, takes place under different conditions. A filament Mlml (fig. 49) in the upper part of the orikice issues with a head h' which may have any value between ir;-Tf;;:Ef»;;;= § ¥ "i'i*"" "°"""" ?:€ E§ ;€gZ-Z-;-tsl;

lr, lim. it

E § ' if i,

1 1 -tl 5 t

- n | 1

5 5 "L " 'hm

M' ' t . UJ i.

I 4, L e- ~= .=- r'E

I mg

-, %.—.. — FIG.

49.

hi and h. But a filament Mmzz issuing in the lower part of the orifice has a velocity due to h”-lt”', or h, simply. In the upper part of the orifice the head is variable, in the lower constant. If Q1, Q2 are the discharges from the upper and lower parts of the orifice, b the width of the orifice, then

Q1=§ cb/ @{h3-hi3} ()

Q.=@1»<1»2-h>~/it ' 3

In the case of a rectangular notch or weir, h1=0. Inserting this value, and adding the two portions of the discharge together, we get for a drowned weir

Q =Cb/ 2£h(h2-h/3), (4)

where h is the difference of level of the head and tail water, and hz the head from the free surface above the weir to the weir crest g. 50 .

From some experiments by Messrs A. Fteley and F. P. Stearns can be reduced

(Trans. Arn. Soc. C.E., 1883, p. 102) some values of the coefficient c ha/hz c' ha/hz c

0-1 o-629 o-7 0-578

0-2 0-614 o-8 0-583

o-3 0-600 0-9 0-596

0'4 0'590 0'95 0'607

0-5 0-582 1-oo 0-628

0-6 0-578

If velocity of approach isitaken into account, let B be the head due to that velocity; then, adding f) to each of the heads in the equations (3), and reducing, we get for a weir W Q=cb-J3§ l(h2+5)(h-I-YJ);-%(h+f>)%-iflgl; (5) an eq uation which may be useful in estimating flood discharges Bridge Piers and other Obstructions in Streams.-When the piers of a bridge are erected in a stream they create an obstruction to the flow of the stream, which

causes a difference of surface-, :;“:;'~j -3 i;~ t ' ' ' “g level above and below the ' " ' ' , pier-(fig. 51). If it is neces- /7 sary to estimate this difference

of level, the flow

between the piers may be

treated as if it occurred over,

a drowned weir. But the Q1

value of c in this case is -V/l0?QJZ//Q/}7J, U%AL//A/Z///Q/A517//*W imperfectly known.

§ 47. Bazin's Researches on

Weirs.-H. Bazin has executed a long series of researches on the flow over weirs, so systematic and complete that they almost supersede other observations. The account of them is contained in a series of papers in the Annales des Pants et Chaussées (October 1888, January 1890, November 1891, February 1894, December 1896, 2nd trimestre 1898). Only a very abbreviated account can be given here. The general plan of the experiments was to establish first the coefficients of discharge for a standard weir without end contractions; next to establish weirs of other types in series with the standard weir on a channel with steady flow, to compare the observed heads on the different weirs and to determine their coefficients from the discharge computed at the standard weir. A channel was constructed parallel to the Canal de Bourgogne, taking water from it through three sluices 0'3XI'O metres. The water enters a masonry chamber 15 metres long by 4 metres wide where it is stilled and passes into the canal at the end of which is the standard weir. The canal has a length of 15 metres, a width of 2 metres and a depth of 1-6 metres. From I '~;;1-

—L — FIG.

50.

"§ if

fi. %¢ * -1 1-7

| V

FIG. 51.11

this extends a channel 200 metres in length with a slope of I mm. per metre. The channel is 2 metres wide with vertical sides. The channels were constructed of concrete rendered with cement. The water levels were taken in chambers constructed near the canal, by fioats actuating an index on a dial. Hook gauges were used in determining the heads on the weirs. Standard Weir.-The weir crest was 3-72 ft. above the bottom of the canal and formed by a plate i in. thick. It was sharp-edged with free overfall. It was as wide as the canal so that end contractions were suppressed, and enlargements were formed below the crest to admit air under the water sheet. The channel below the weir was used as a gauging tank. Caugings were made with the weir 2 metres in length and afterwards with the weir reduced to 1 metre and 0-5 metre in length, the end contractions being suppressed in all eases. Assuming the general formula Q = mlhwl (2gh), (I) Bazin arrives at the following values of ma-

Coejicients of Discharge of Standard Weir. Head h metres. Head h feet. m 1 0-05 -164 0-4485 o-10 -328 0-4336 § 0-15 '492 0-4284 Q 0-20 -656 0-4262 1 0-25 '820 0-4259 § 0-30 -984 0-4266 0-35 1-148 0-4275, 0-40 1-312 0-4286 0'45 1-476 0'4299 . 0-50 1-640 0-4313 0'55 V304 0'4327 0-60 1-968 0-4341 I Bazin compares his results with those of Ftele and Stearns in 18 Y 77 and 1879, correcting for a different velocity of approach, and finds a close agreement. I nfluence of Velocity of A pproach.-To take account of the velocity of approach it it is usual to replace h in the formula by h-I-au*/2g where a is a coefficient not very well ascertained. Then Q =#l(h+<»1¢'/2g)~/ l2g(h+au”/2g)} =#lh' (2§ h)(1 +fw2/2gh)3~ (2) The original simple equation can be used if m =;i(1 -l-au”/2gh)% or very approximately, since 142/2gh is small, m =/-(I +%-w”/2gh)- (3) Now if p is the height of the weir crest above the bottom of' the canal (fig. 52), u=Q/l(p-l-h). ""~ ~—. ' Replacing Q by its value L i in (1) rf—'-=Q2¢zrz-;t.~;@>”* H

m

"¢i"" P x“.: so that (3) may be written 4

'ge, ia s m=~u+/-th/<z>+h>s21. <0

Gaugin s were made with y/ yy/W//A///M weirs 0? 0-75, 0-50, 0-35, and 0-24 metres height above the canal bottom and the results compared with those of the standard weir taken at the same time. The discussion of the results leads to the following values of m in the general equation (1):- m =+-(1 +2-su'/2gh) Fig. 52. from the weir, but encloses a volume of air which is at less than atmospheric pressure, and the tail water rises under the sheet. The discharge is a little greater than for free overfall. At greater head the air disappears from below the sheet and the sheet is said to be “ drowned." The drowned sheet may be independent of the tail water level or influenced by it. In the former case the fall is followed by a rapid, terminating in a standing wave. In the latter case when the foot of the sheet is drowned the level ff 1'4'3'~"~3= ii=;; of the tail water influences fl *:Q;*>f'~the discharge even if it is "'“"" " '°-Q5 below the weir crest. 'j Weirs with Flat Crests.~ F A The water sheet may spring °' ”"' "" clear from the upstream edge 1 Y A of may adhere to the fiaf WW WW/A% crest falling free beyond the FK; 53 downstream edge. In the former case the condition is that of a sharp-edged weir and it is realized when the head is at least double the width of crest. It may arise if the head is at least 1% the width of crest. Between these limits the condition of the sheet is unstable. When the sheet is adherent the coefficient m depends on the ratio of the head h to the width of crest e (fig. 53), and is given by the equation m=m1 [0-70-i-0-185h/c], where ml is the coefficient for a sharp edged weir in similar conditions. Rounding the upstream edge even to a small extent modifies the dis- ' ' " ' 'lcharge. If R is the radius of the rounding the coefficient m is increased in the ratio 1 to 1 +R/h nearly. P The results are limited to R less than é in. h ' "'~v.

'~ ..c

}— - . Drowned Weirs.~Let h WW//#yy/M///5 (fig. 54) be the height of FIG head water and hi t at of ' 54 tail water above the weir crest. Then Bazin obtains as the approximate formula for the coefficient of discharge fl = I '05m1lI +%h1/1>li/ lfh '° hx)/hi, where as before ml is the coefficient lor a sharp-edged weir in similar conditions, that is, when the sheet is free and the weir of the same height. § 48. Separating Weirs. - Many towns derive their

"';"': {e-t-:¢;-*é f § // ' =u[I +0°55lh/(l>+h)l”l-Values of p- Head h metres. Head h feet. p 0-05 -164 o-4481 0-10 -328 0'4322 0-20 -656 Q°4215 0'30 '984 0'4174 0-40 1-312 0-4144 0-50 I'640 0-4118 0-60 1-968 0:4092 V An approximate formula for it is: /l=O'405~i-0-003/h (h in metres) / fu . f Z %1 § ' § /, s s water supply from < < - < . : "' “ streams in high moorland districts, in which the flow is extremely variable. The water is collected in large storage reservoirs, from which an uniform supply can be sent to the town. In ff? si e '§ “§ s § FIG. 55. Plan, of Ccwt Iron Key /4=o-405-{-0-01/h (h in feet). I nelined Weirs.eExperiments were made in which the plank weir was inclined up or down stream, the crest being sharp and the end contraction suppressed. The following are coefficients by which the discharge of a vertical weir should be multiplied to obtain the discharge of the inclined weir. Coefficient. Inclinauon up stream . 1 to 1 0-93 n sy 3 to 2 3 to 1 0-96 Vertical weir .... . . 1-00 Inclination down stream . 3 to 1 1-04 3 to 2 1-07 1 to 1 1~10 1 to 2 1-12 I to 4 1-09 The coefficient varies appreciably, if h/p approaches unity, which case should be avoided. In all the preceding cases the sheet passing over the weir is detached completely from the weir and its under-surface is subject to atmospheric pressure. These conditions permit the most exact determination of the coefficient of discharge. If the sides of the canal below the weir are not so arranged as to permit the access of air under the sheet, the phenomena are more complicated. So long as the head does not exceed a certain limit the sheet is detached 5%//¢'r;' //¢/;;<e/f;/tgegg.;

9"=' 5 =;5'f-3'f¢'i'~§ 3?<f'~'3»3Z1»~3“ '¢ f ¢=zs3e.==r<ss~.-¢¢?2t€'¢>;ias-fi-ef'/% sggfgr.-if g;~'1.°f.;9r~1;»a=.-feafw V. /7/ 4 as:—Arsf if.-x-f.="»§ t>= .9 ff W/ /A why .R-1-"H-: '< ” "@§ < 1' 1 f f"" war LKQQ, 7 ¢.'.. Edgy /V Li .1 wi-44, L- ~ - '1. ”;, .=;— f /', N//7 ~//" “' - %"f@'?r,

150/////;, ,, , 57' fx '/ '/ /4, .Z A, T/Q" 1-7 f// W/A %% ///” %//5 / /» - ~ / /4./3/.mn / M * i:»'°;'L.§ N 'F *csv* V f 1. ...s-..<=.;; -=~ . ///f/, .>. azrfw 11,1-y.q°, ;':£-T.; V Tr; ix* - ':ie"G£; 7 . - ”, /4.=':»;;{;;, g “ »» 'J 'r. Vi.. ~ 1 J <:MU 'r -8 C., ».. 'Y Q: =& Q-, sgfysg-Q2-:ig ~, > .- = ..<- s.e.° § ", -UH -

W Z/ 2 ~, ,, -, '° fr; 'nh XY /tl 9 / / / / '°., e§ ¢.' 5//f @/”/// > ', 1. 1” / 1/ // / / f/ / /., / ge, ” . — /5°-°, , / /i / /

H ¢-vii; 5/ / ////Q/ § / 'Q /':bf-H" sr/// /9 /'/'/

/ °.-sw / / / 5 5

§ "° ' //A 2 3.

U, ” 'l W /7 /, »°:", § f,

W c/ / .es

  • , '§ 1'////Q /.fe es

pw »¢=¢-, ,~/ / //, as ° //{, -»s'..s¢, Q /s, ,{) > - .- L . /, f, - e, //. ., -. me-“;- “=“..?, ,.' .f..;'§ >;/ //~ / -///., /I 1 f/ /, y, -;f-¢“£'e, .i-0. m~'— .M wg-.r =...'-»1a, / ////-' /, , f .~».», :, »;, - .e-;°qsif

  • »'“%s.f. 2? !fl '-, // /6 /Qi//, j~*>»-»;-5 vi(

/ of .~ = ff' A / fs* Wi" / / '§ $~"' ' SEM/ //'”/f/ //4 V A "'; [;LQ*@i;.: 142.-'J 5.6/Ze, / /°» Q-.', :1";.~s . Ts.;..="¥»l'lWg/I § “, ~Ig2-i5 .f:i'¢-1-1-". .';.—f:="*3:=?' .gin-* ~ *J-'“¢.>l 'i'€S ww .ws ":~1~*'.'f'=¥rt~1?° ~'-2'<€°"'>¥ he -e=7-*"-'silos, —fr, -r-°$*r:~, .t2 “Ff

  • .s—J~'°=;-:-+1 {»*f;~'§ .=Sf-:~;:¢f~° r”.¢~s~ ~;~. a;'~>'3ie—www 1 'l“"l~1

-cs -i1§ >'r<»e'°-.J .t ~,1¢~=»=.~, $v ~:':, 'a;'.-' -'.=fg.». »- -fi . ~ ~ - . —: 'J-f- wni-s 'I°-2..—.¢r—f'“~"—f=-=§ ?~.- f'-fm 5-“;%;i-; |, ,,5¥-b Rr mtg;-515-5-, fi-F:ss-}%1::C "'~':'Fft'§ :, ;§ L'53<§ 2: Agrffa m¥, mfh5' 3, ;, 'a;&g, emo s metre e, L.r...;, , w¢, ,~ 13, ., . .g FIG. 56. such cases it is desirable to separate the coloured water which comes down the streams 1n high Hoods from the purer water of ordinary flow. The latter is sent into the reservoirs; the former is allowed to flow away down the original stream channel, or is stored in separate reservoirs and used as compensation water. To accomplish the separation of the flood and ordinary water, advantage is taken of the different horizontal range of the parabolic path of the water falling over a weir, as the depth on the Weir and, consequently, the velocity change. Fig. 55 shows one of these separating wcirs in the form in which they were first introduced on the Manchester Waterworks; fig. 56 a more modern weir of the same kind designed by Sir A. Binnie for the Bradford Waterworks. When the quantity of water coming down the stream is not excessive, it drops over the weir into a transverse channel leading to the reservoirs. In flood, the water springs over the mouth of this channel and is led into a waste channel. it may be assumed, probably with accuracy enough for practical purposes, that the particles describe the parabolas due to the mean velocity of the water passing over the weir, that is, to a velocity § / (2g71), where iz is the head above the crest of the weir. Let cb=x be the width of the orifice and ac=y the difference of level of its edges (fig. 57). Then, if a particle passes from a, to b in t secon s, y=%g¢', x=§ ~/ (2gh)¢; 3'= 1311962/fl, which gives the width x for any given difference of level y and head h, which the jet will just pass over the orifice. Set off ad vertically 1 ~ 11 ¢ /

QW/ f /, . fy,

t//, 5: yi 5: l it I 1 | | I 1 I | v I 1 1 1 1 1 - I I I E 1 I 2 I % E

-

f Q T . >, -

0 r —~—n ' . I <~-...i°;; ., A> 1,7 < .

 ' .

1-:<jb, Z. , .Y, M

.4, , >, ? -i'~.

,

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 , ' Q .
.
.
  P(-
 .;
 '; 's
1 1 5


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.
f


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9 i, 3 e

Fig. 57. and equal to ég on any scale; af horizontally and equal to § / (gh). Divide af, fe into an equal number of equal parts. Join a with the divisions on ef. The intersections of these lines with verticals from the divisions on af give the parabolic path of the jet. MOUTHPIECES-HEAD CONSTANT § 49. Cylindrical Ivloulhpieces.-'Vhen water issues from. a short cylindrical pipe or mouthpiece of a length at least equal to 1% times its smallest transverse dimension, the stream, after contraction within the mouthpiece, expands to fill it and issues full bore, or without contraction, at the point of discharge. The discharge is found to be about one-third greater than that from a simple orifice of the same size. On the other hand, the energy of the fluid per unit of weight is less than that of the stream from a simple orifice with the same head, because part of the energy is wasted in eddies produced at the point where the stream expands to fill the mouthpiece, the action being something like that which occurs at an abrupt change of section. Let fig. 58 represent a vessel discharging through a cylindrical mouthpiece at the depth h from the free surface, and let the axis of the jet XX be taken as the datum with reference to which the head. is estimated. Let Q be the area of the mouthpiece, w the area of the stream at the contracted section EF. Let 'v, p be the velocity and pressure at EF, and vi, pt the same quantities at GH. If the discharge is into the air, pl is equal to the atmospheric pressure pa. The total head of any filament which goes to form the jet, taken at a point where its velocity is sensibly zero, is IH-pa/G; at EF the total head is v2/2g-l-p/G; at GH it is 1112/2g-l-pl/G. Between EF and GH there is a loss of head due to abrupt change of velocity, which from eq. (3), § 36, may have the value (ff-v)2/2g-Adding this head lost to the head lat GH, before equating it to the heads at EF and at the point where the filaments start into motion, - h~-Pa/G =v2/Qgefp/Q1 =f»12/2g+¢>1/G+<»-vm/Qg, But wv=Slv1, and w =c, § l, if 0, is the coefficient of contraction within the mouthpiece. Hence 1/=§ Z'Ui/w=v1/cc. Supposmg the discharge into the air, so that pl =p., , h+t>@/G =v1°./2§ '+1>a/G+(vi2/2g> (I/cr-I)': (19/22) ll +*}=h: v1= v<2gh>/v:1+<1/U.-1>2x; <1> where the coefficient on the right is evidently the coefficient of velocity for the cylindrical mouthpiece in terms of * T2: the coefficient of contraction at EF. Let T':ii=:'lL.".l.';"' .l 6¢=0~64, the value for '”"“a"' *“** simple orifices, then the { coefficient of velocity is ' it cv=1//{1+(1/ce-1)”l =<>-81 <2>, C G F I f:, ';;a§ ;f.;1L€;§§ . x t >> 1 0-82, which does not / ~ differ more from the f theoretical value than / might be expected ifthe friction of the FIG 58 mouthpiece is allowed for. Hence, for mouthpieces of this kind, and for the section at GH, ¢, ,=o~82 ce = I'OO C=O'82, Q =o-82524 (zgh). It is easy to see from the equations that the pressure 12 at EF is less than atmospheric ressure. Eliminating vt, we get <'§ 3.- p>/Q = ih nearly; e> or p=p, ,-§ (, ~hTb per sq. ft. If a pipe connected with a reservoir on a lower level is introduced into the mouthpiece at the part where the contraction is formed (fig. 59), the water will rise in this (pipe to a height KL == (pa -p)/ = in nearly. If the distance X is less than this, the water from the lower reservoir will be forced continuously into the jet by the atmospheric pressure, and discharged with it. This is the crudest form of a kind of pump known as the jet pump. § 50. Convergent Illouthp/ieces.—With convergent mouthpieces there is a contraction within the mouthpiece causing a loss of head, and a diminution of the velocity of discharge, as with cylindrical mouthpieces. There is also a second contraction of the stream outside the mouthpiece. Hence the discharge is given by an equation of the form Q =5vCcQ/ (2gh)v where KZ is the area of the external end of the mouthpiece, and 6,9 the section of the contracted jet beyond the mouthpiece. Convergent Mouthpieces (Castel's Experiments).-Smallest diameter of orzjice =o-o5o85 ft. Length of mouthpiece=2-6 Diarneters. A I f Coefficient of Coefficient Coefficient of C ng 6 0 Contraction, Velocity, Discharge, 0IlV€Yg€1'lC€. CE Cv C o° o' '999 -830 ~S29 1° 36' 1-000 -866 -866 3° 10' I-001 -894 895 4° 10' I'O02 'QIO 912 5° 26' 1-004 -920 924 gi sgf -993 -gi; 329 5 '99 ' 34 10° zo' '987 ~950 -938 12° 4' 986 '955 942 “Z 25 333 322 94° I4 2 - 941 162 36' 959 '971 ' 938 19° 23, 953 '970 '924 21° of 945 -971 918 23 0 937 '974 913 29° 58' '919 -975 -896 40° 20' 887 ~980 869 48° 50' 861 -984 847 The maximum coefficient of discharge is that for a mouthpiece with a convergence of 13° 24'. The values of cf and cs must here be determined by experiment. The above table gives values sufficient for practical purposes. Since I the contraction beyond the mouthpiece increases -E 1.:;;: . - Tgl with the ionvergenicr, what is the same t ing, N! Zi 66 diminishes, and on the 5 other hand the loss of fi I energy diminishes, so Q Q that Uv increases with Q J the convergence, there “ Q 2;:fjj;;g is an angle for which the § -- ', . product c¢ cv, and con/ N sequently the discharge, I -is a maximum. |= K 5 § 5I. Divergent Corz)f oidal Mouthpiice.-Sup- pose a mout piece so I designed that there is - Q no abrupt change in the 1. L, , l[ section or velocity of "" the stream passing through it. It may A have a form at the inner end approximately the same as that of a simple contracted vein, and may then enlarge gradually, as shown in fig. 60. Suppose that at EF it becomes cylindrical, so that the jet may be taken to be of the diameter EF. Let av, t', p be the section, velocity and pressure at CD, and Q, 1/1, pl the same quantities at EF, pn being as usual the atmospheric pressure, or pressure on the free surface AB. Then, FIG. 59. DISCHARGE WITH VARYING HEAD A B


since there is no loss of energy, except the small frictional resistance of the surface of the mouthpiece, h-I-p.1/G=v2/2g-l-p/G —|— =v12/2g-l-pl/G. Q Ili; the jet discharges into I t eair, p1=p.1;an 1 1/12/2g=h; Zi vi = / (2gh); 5 or, if a coefficient is introduced to allow for friction, ic E “vi =cv/ (2gh): 2 | where Cv is about o~97 if I the mouthpiece is smooth 5' ' 5 and well formed. » ~'—*- '*-"- i'-“ "'-'*"" ' Q = S2211 =6vQ/ (2gh). Xf' Hence the discharge depends on the area of the stream at EF, and not at ID F all on that at CD, and the latter may be made as small as we please without FIG 60 affecting the amount of water discharged. There is, however, a limit to this. As the velocity at CD is greater than at EF the pressure is less, and therefore le ss than atmospheric pressure, if the discharge is into the air. If CD is so contracted that p=o, the continuity of flow is impossible. In fact the stream disengages itself from the ° mouthpiece for some value of p greatir than o (fig. 61). From the equations, / P/f' =1>f»/G * (vi -11 2)/22° l WV Let il/w =m. Thenl U = 7] m; s P/G=pa/Giv.2<m2-1>/2g

=Pu/G-(Wiz-1)}l;

" ' whence we find that p/G $

will become zero or negative

~ Q/~ § ~{1<h+1>./G>/hz = /11 +Pa/Ghf; FIG 5I or, putting pa/G=34 ft., if S2/w§ ~/{(h-l-34)/h}, In practice there will be an interruption of the full bore How with a less ratio of Q/cv, because of the disengagement of air from the water. But, supposing this does not occur, the maximum discharge of a mouthpiece of this kind is Q =w~' f2g(h+Pa/GN: that is, the discharge is the same as for a well-bell mouthed mouthpiece of area w, and without the expanding part, discharging into a vacuum. § 52. Jet Pump.-A divergent mouthpiece may be arr angled to act as a pump, as shown in hg. 62. The water which supplies the energy required for pumping enters at A. The water to be pumped enters DD where the velocity is greatest DD the stream enlarges in section, at B. The streams combine at and the pressure least. Beyond . 6" ~. // f mM/ 'af

/ / g 3 EQTQ
 f

% // f /' /0 / 'S s)§ / //, § gg£f M 5 .,

.L gg; - f p s § T * '< <1s. 駠;?§ ?;

E§ " , . H el; =, / / f-, sge gas. as ~< -§ s- " = / /, /. /V t s =g Mk, M //, I / i s D i' / f / .», '/, ,¢ 'mls / / r I if %>s~ Y/ / nhl / 7/sg N/ l;' Z // / J li* I //§ fi /M o / /, <>=®3 >.§ T;§ fj';?-Jlfgill* at i/ if g/ $54 » il [1 li ;Z” hi, s i ii g

fi. lilli . fi ll I U

A r and its pressure increases, till it is sufficient to balance the h ead due to the height of the lift, and the water Hows away by the discharge pipe C. FIG. 63 shows the whole arrangement in a diagrammatic way. A is the reservoir which supplies the water that effects the pumping; "" ' "' "" """"'"""' ' " 7 5§ :§ ?'§§ 71 ia;-':vi'E=.; if I """!“"" I I a if i) 1? - fK - .. - - -~ .; -;-. - -1——~ .: a J. Q QE 1 D D f”2 l? I B v .i» FIG. 63. B is the reservoir of water to be pumped; C is the reservoir into which the water is pumped. § 53. Flow from a Vessel when the Effective Head 'varies with the Time.—Various useful problems arise relating to the time of emptying and filling vessels, reservoirs, lock chambers, &c., where the How is dependent on a head which increases or diminishes during the operation. The simplest of these problems is the case of filling or emptying a vessel of constant horizontal section. Time of Emptyirzg or Filling a Verlical-sided Lock Chamber.-Suppose the lock chamber, which has a water surface of Sl square ft., is emptied through a sluice in the tail gates, of area ai, placed below the tail-water level. Then the effective head producing flow through the sluice is the difference of level in the chamber and tail bay. Let H (fig. 64) be the initial difference of level, h the difference =;E?-E15-;§ “1§ —. - -C»—-H dh A 1, , x . I It 1 i Tuul water lamb 3»;§ a~;T~§ e§§ >- -ésezf

F IG. 64. of level after l seconds. Let -dh be the fall of level in the chamber during an interval dt. Then in the time dt the volume in the chamber is altered by the amount -Sldh, and the outfiow from the sluice in the same time is cw/ (2gh)dt. Hence the differential equation connecting h and t is caul (2gh)dt-i-Qh =(J. For the time t, during which the initial head H diminishes to any other value h, -in/(Cm/2g>} hdh//h= tdt. H 0 -'~t=29(/H -/h)/iCw1/ (2£)i = (9/¢w){~/ (2H/g) -~/ (2h/g)}-For the whole time of emptying, during which h diminishes from T = (Sl/eww (2H/g)-Comparing this with the equation for flow under a constant head, it will be seen that the time is double that required for the discharge of an equal volume under a constant head. The time of filling the lock through a sluice in the head gates is exactly the same, if the sluice is below the tail-water level. But if the sluice is above the tail-water level, then the head is constant till the level of the sluice is reached, and afterwards it diminishes with the time. PRACTICAL Use or OR1r1CEs IN GAUGING WATER § 54. If the water to be measured is passed through a known orifice under an arrangement by which the constancy of the head is ensured, the amount which passes in a given time can be ascertained by the formulae already given. It will obviously be best to make the orifices of the forms for which the coefficients are most accurately determined; hence sharp-edged orifices or notches are most commonly used. Waler I nch.-For measuring small quantities of water circular sharp-edged orifices have been used. The discharge from a circular orifice one French inch in diameter, with a head of one line above the top edge, was termed by the older hydraulic writers a water-inch. A common estimate of its value was 14 pints per minute, or 677 English cub. ft. in 24 hours. An experiment by C. Bossut gave 634 cub. ft. in 24 hours (see Navier's edition of Belidofs Arch. Hydr., p. 212). L. ]. Weisbach points out that measurements of this kind would be made more accurately with a greater head over the orifice, and he proposes that the head should be equal to the diameter of the orifice. Several equal orifices may be used for larger discharges. Pin Ferrules or Measuri1zg Cocks.-To give a tolerably definite supply of water to houses, without the expense of a meter, a ferrule with an orifice of a definite size, or a cock, is introduced in the service-pipe. If the head in the water main is constant, then a definite quantity of water would be delivered in a given time. The arrangement is not a very satisfactory one, and acts chiefly as a check on extravagant use of water. It is interesting here chiefly as an example of regulation of discharge by means of an orifice. Fig. 65 shows a cock of this kind used at Zurich. It consists of three cocks, the middle one havin Htoo, g§ , » Q ll. 12%;} f <~ Q /7 T' &:> ?é§ is ® Q < A § A mminmumxmnn /" mmm n nu Z nv ummm: un my § '§ -, %i'§ § . Q / gi' - “l ll! FIG. 65. g the orifice of the predetermined size in a small circular plate, protected by wire gauze from stoppage by impurities in the water. The cock on the right hand can be used by the consumer for emptying the pipes. The one on the left and the measuring cock are Connected by a key which can be locked by a padlock, which is under the control of the water company. § 55. M'measurement of the Flow in Streams.-To determine the quantity of water flowing off the ground in small streams, which is available for water supply or for obtaining water power, small temporary weirs are often used. These may be formed of planks supported by piles and puddled to prevent leakage. The measurement of the head may be made by a thin-edged scale at a short distance behind the weir, where the water surface has not begun to slope down to the weir and where the velocity of approach is not high. The measurements are conveniently made from a short pile driven into the bed of the river, accurately level with the crest of the weir (fig. 66). Then if at any moment the head is h, the discharge is, for a rectangular notch of breadth b, Q = § cbh/ 2gh where c=0-62; or, better, the formula in § 42 may be used. Gauging weirs are most commonly in the form of rectangular notches; and care should be taken that the crest is accurately horizontal, and that the weir is normal to the direction of flow of the stream. If the planks are thick, they should be bevelled (fig. 67), and then the edge may be protected by a metal plate about fifth in . thick to secure the requisite accuracy of fo1'm and sharpness of edge. In permanent gauging weirs, a cast steel plate is sometimes used to form the edge of the weir crest. The weir should be large enough to discharge the maximum volume flowing in the stream, and at the same time it is desirable that the minimum head should not be too small (say half a foot) to decrease the effects of errors of measurement. The section of the jet over the weir should not exceed one-fifth the section of the stream behind the weir, or the velocity of approach will need to be taken into account. A triangular notch f l |

I 1S very suitable for measurements of this kind. If the How is variable, the head h must be recorded at equidistant intervals of time, say twice daily, and then for each I2-hour period f Scale Wet r L.; .;1 4.13:;:¢; -t ~:.i -e3$§ -'- »-= -'Q—"' ': / I /Q —a -~ .».§ §§ =f:;/;=1 is =~<§ 1 ¢w1 i .r-'a- -'wary-1w1.., ,. ff: Ev:;;'7-'731: —::::°., f -'., 1 .. 1. I., ,, V - Q gy § &-f, :, ,, ,?, ,Zm, %, ¢., ,, , ~ 2 f Z f I i C

FIG. 66. the discharge must be calculated for the mean of the heads at the beginning and end of the time. As this involves a good deal of troublesome calculation, E. Sang proposed to use a scale so graduated as to read off the discharge in cubic feet per second. The lengths of the principal graduations of such a scale are easily calculated by putting Q=r, 2, 5 . in the ordinary formulae for notches; the intermediate graduations may be taken accurately enough by subdividing equally the distances between the principal graduations. The accurate measurement of the discharge of a stream by means of a weir is, however, in practice, rather more difficult than might be inferred from the simplicity of the principle of the /, -T operation. A art 6; from the difficiilty Z* of selecting a suitable coefficient of W discharge, which need not be serious if the form of the j, weir and the nature of ilts crest aire propery atten ed to, other difficulties of FIG' 67 measurement arise. weir should be very accurately determined, and if the weir is rectangular its deviations from exactness of 'level should be tested. Then the agitation of the water, the ripple on its surface, The length of the and the adhesion of the water to the scale on which the head is measured, are liable to introduce errors. Upon a weir IO ft. long, with 1 ft. depth of water fiowing over, an error of Hoooth of a foot in measuring the "head, or an error of Hooth of a foot in measuring the length of the weir, would cause an error in computing the discharge of 2 cub. ft. per minute.-Hook Gauge.-For the determination of the surface level of water, the most accurate instrument is the hook gauge used first by U. Boyden of Boston, in 1840. It consists of a fixed frame with scale and vernier. In the instrument in fig. 68 the vernier is fixed to the frame, and the scale slides vertically. The scale carries at its lower end a hook with a fine point, and the scale can be raised or lowered by a fine pitched screw. If the hook is depressed below -1 - *-1 S 3. o 1; EE fl e 1 2 é I e I e - f

| 2

l' 9 1 is @| ' é

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E 2 I 4 I E | é 3; t (D i g | 2 2 I é

Q l g 1' 2 I; §

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E; -; Fig. 68 the'water surface and then raised by the screw, the moment of its reaching the water surface will be very distinctly marked, by the reflection from a small capillary elevation of the water surface over the point of the hook. In ordinary light, differences of level of the water of -001 of a foot are easily detected by the hook gauge. If such

a gauge is used to determine the heads at a weir, the hook should

Canal from the irrigation channel, arrangements termed modules are used. These are constructions intended to maintain a constant or approximately constant head above an orifice of fixed size, or to regulate the size of the orifice so as to give a constant discharge, notwithstanding the variation of level in the irrigating channel.

§ 57. Italtan Module.-The Italian modules are masonry constructions, consisting of a regulating chamber, to which water is admitted by an adjustable sluice from the canal. At the other end of the chamber is an orifice in a thin flagstone of fixed size. By means of the adjustable sluice a tolerably constant head above the fixed orifice is maintained, and therefore there is a nearly constant discharge of ascertainable amount through the orifice, into the channel leadinfg to the fields which are to be irrigated. In g. 69, A is the adjustable sluice by which water is admitted to the regulating chamber, B is the fixed orifice through which the water is discharged. The sluice A is adjusted from time to time by the canal officers, so as to bring the level of the water in the regulating chamber to a fixed level marked on the wall of the chamber. When qw 32 it 1 2: f/ /; I I " r////////////////H///////////W l time to time. It has further the advantage that the cultivator, by observing the level of the water in the chamber, can always see whether or not he is receiving the proper quantity of water. On each canal the orifices are of the same height, and intended to work with the same normal head, the width of the orifices being varied to suit the demand for water. The unit of discharge varies on different canals, being fixed in each case by legal arrangements. Thus on the Canal Lodi the unit of discharge or one module of water lS the discharge through an orihce 1-12 ft. high, 0-12416 ft. wide, with a head of o-32 ft. above the top edge of the orifice, or -88 ft. above the centre. This corresponds to a discharge of about o-6165 cub. ft. per second. In the most elaborate Italian modules the regulating chamber is arched over, and its dimensions are very exactly prescribed. Thus l, in the modules of the Naviglio Grande of Milan, shown in fig- 70, l l I the measuring orifice is cut in a thin stone slab, and so placed that the discharge is into the air with free contraction on all sides. The - LL c, , i, E-'; = -:it=, -:='° - - '~*:- —'i;7'“ "';:" rf~ UF* ” | O | 2 I I I g ~§ s , ” Y..... § sf- ~— § , — '-1Z Slowking4Farmbrough's revenge*'l'@"~I'4 FIG. 71. adjusting sluice is placed with its sill flush with the bottom of the canal, and is provided with a rack and lever and locking arrangement. The covered regulating chamber is about 20 ft. long, with a breadth I-64 ft. greater than that of the discharging orifice. At precisely the normal level of the water in the regulating chamber, there is a ceiling of planks intended to still the agitation of the water. A block of stone serves to indicate the normal level of the water in the chamber. The water is discharged into an open channel O-655 ft. wider than the orifice, splaying out till it is I-637 ft. wider than the orifice, and about 18 ft. in length. § 58. Spanish Module.-On the canal of Isabella II., which supplies water to Madrid a module much more perfect in principle than the Italian modulo is employed. Part of the water is supplied for irrigation, and as it is very valuable its Tia? .j V ' 'l;2 INC strict measurement is essential. The

j - ¢ // ~ %/Z 'l i I I l I l . | I l R " jl ' 1 I 1 ' i ' 1 U I 5 i 1 | l-1 l ' ' ' Qi

g I I I I ig
5 I J §

| " M? IE l i I l ' I if#, 7 "mf /M / f 1 »i ~ / f/ ff / fr f / r /f W///W//ww %m /M / / r / wr 'S $23! W%/M/%WWWW /WA%/ W/ / //', W ”f@”W€”'/7/W” gf"

ai Y” I module (fig. 72) consists of t 1;-3 a; ' of V chambers one above the other, th; Dy, -I -I 'JI? ~ upper chamber being 1n free communi§ ~3;f, //M VU/ //f@V4é'V'7/W//WW(W-% // . 2 . f, . cation with the irrigation canal, and gwmg f » f . . l/% W %%/V/V/W%, th? lower Cgamgig discharging by a cuverttote e s. lnthearhd roof between the chambers there Cisea / Circular illiarp-edged orgice in a bronze f p ate. anging in t is th e ' a Q ///7 // broiézeé plug of vaiiiablebdiamgierlgus/, , 5 17' 'fW// ' / f h ' . If % W / §§§ , ;;.;f;2..r...°..';“C..f;fi..f:;;, ti. / ass-> Q ' plug descen(tfs and giyes arnh enlarged r ' .1 / I, ,, , A opening, an conyerse y. us a per/

./I  »   fectly constant discharge with a vary~,

”' / /f / " % ing head can be obtained, provided no 4 / FIG. 70.-Scale pkg. ' % clogging or silting of the chambers pre/ vents the free discharge of the Water adjusted it is locked. Let wr be the area of the l or the rise and fall of the float. The theory of the module is very / orifice through the sluice at A, and W2 that of the simple. Let R (fig. 71) be the radius of the fixed opening, r the fixed orifice at B; let hi be the difference of level radius of the plug at a distance h from the plane of flotation of the / between the surface of the water in the canal and ta g . pe since the flow through the orifices at A and B is the same, Q =~'Ciw1/ (22111) =C2w2/ (2Zhf2)» where cr and Q are the coefficients of discharge suitable for the two orifices. Hence regulating chamber; hg the head above the centre of the discharging orifice, when the sluice has been adjusted and the flow has become steady; Q the normal dischar e in cubic feet r second. Then, Cl(/J1{C2C02 '-= / If the orifice at B opened directly into the canal without any intermediate regulating chamber, the discharge would increase for a given change of level in the canal in exactly the same ratio. Consequently the Italian module in no way moderates the fluctuations of discharge, except so far as it affords means of easy adjustment from float, and Q the required discharge of the module. Then Q =c1r(R°'-r2)/ (zgh). Taking c =o-63, Q = 15-88(R”-r2)/ h; r=/{R'-Q/I5~88/h}. I Choosing a value for R, successive values of r can be found for different values of h, and from these the curve of the plug can be drawn. The module shown in fig. 72 will discharge I cubic metre per second. The fixed opening is o-2 metre diameter, and the greatest head above the fixed orifice is I metre. The use of this module involves a great sacrifice of level between the canal and the fields. The module is described in Sir C. Scott-Moncrieff's Irrigation in Southern Europe. § 59. Reservoir Gauging Basins.-In obtaining the power to store the water of streams in reservoirs, it is usual to concede to riparian owners below the reservoirs a right to a regulated supply throughout the year. This compensation water requires to be measured in such a way that the mill owners and others interested in the matter can assure themselves that they are receiving a proper quantity, and they are generally allowed a certain amount of control as to the

FIG. 72. g, ,', ,, Fig. 74 shows an arrangement designed for the Manchester water works. The water enters from the reservoir a chamber A, the object of which is to still the irregular motion of the water. The admission is regulated by sluices at b, b, b. The water is discharged by orifices or notches at a, a, over which a tolerably constant head is maintained by adjusting the sluices at b, b, b. At any time the mill owners can see whether the discharge is given and whether the proper head is maintained over the orifices. To test at any time the discharge of the orifices, a gauging basin B is provided. The water ordinarily flows over this, without entering it, on a floor of cast-iron plates. If the discharge is to be tested, the water is turned for a definite time into the gauging basin, by suddenly opening and closing a sluice at c. The volume of How can be ascertained from the depth in the gauging chamber. A mechanical arrangement (Hg. 73) was designed for securing an absolutely constant head over the orifices at a, a. The orifices were formed in a cast-iron plate capable of sliding up and sg ' l;;»;s

  • ~ f""*.'

Lgsss 14 / a- f 'a fish iifdlffli é///W//“aw W/ f /4 //'r/ff f2'~?}' if B zé2%%%/2%//54;/éffz/////Q/, ff?»»/<»if/<gg/%@//Z@¢;<¢;/4gficaf/Z2/;Q>;;/, ig;;@<§§ >;,2

i;f§ ;-2 aa/ /A //W /7100/'////0/fy///'//47/'/;£52 f7<f//<7

fd, , z%QiQ%{5%», %//£61/3% if//?//='//?2/245 éff 2/CQ///fr FIG. 73.-Scale Th. down, without sensible leakage, on the face of the wall of the chamber. The orifice plate was attached by a link to a lever, one end of which -rested on the wall and the other on floats f in the chamber A. The floats rose and fell with the changes of level in the chamber, and raised and lowered the orifice plate at the same time. This 1/ TZ Ji 2 Z Z/ ) f/ t 12/ f ¥ '§

TT -, -. sf f », ¢z7&;1<»”- ff" " fy'//7/4, f/, ff//4/f/0/yimfga /< 6 VW L ', ,.

= f/ PLEH. Q' Z, 6 “ zigr a a 5 =gs: = ,

§ , >*'°*, /, , >», <, »w77>/* », /ig, /;;

“//1 77,7/7?/i/7647 will” /GQ/ /gy & 02% - , ,/// ~ ' , /yy // ? ~

c i A B Elms- m g ., W 'W ' so - ¢,7/vfff,

  1. 2==;~=1;f1F~§

g//4/f;¢;/ff/%»;;4w 7 //Z, “&'q}'#f', .6f 5;;, %//, //, ,, /., /Zz-f im im mf 411.:MZ 1 fl f/21/2 m. A /4, /az/», rx/ /////ffm* av//:ffm/%z%¢ Fig. 74.-Scale 555. mechanical arrangement was not finally adopted, careful watching of the sluices at b, b, b, being sufficient to secure a regular discharge. The arrangement is then equivalent to an Italian module, but on a lar e scale. 60. Professor Fleeming ]enkin's Constant Flow Valve.-In the modules thus far described constant discharge is obtained by varying the area of the orifice through which the water Hows. Professor F. Ienkin has contrived a valve in which a constant pressure head is obtained, so that the orifice need not be varied (Roy. Scot. Society of Arts, 1876). Fig. 75 shows a valve of this kind suitable for a 6-in. water main. The water arriving by the main C passes through an equilibrium valve D into the chamber A, and thence through a sluice O, which can be set for any required area of opening, into the discharging main B. The object of the arrangement is to secure a constant difference of pressure between the chambers A and B, so that a constant discharge flows through the stop valve O. The equilibrium valve D is rigidly connected with a plunger P loosely fitted in a diaphragm, separating A from a chamber B2 connected by a pipe B1 with the discharging main B. Any increase of the difference of pressure in A and B will drive the plunger up and close the ||

f I 1

ll 'En

in | |lI, .

FIG. 75.-Scale 91.5.

equilibrium valve, and conversely a decrease of the difference of pressure will cause the descent of the plunger and open the equilibrium valve wider. Thus a constant difference of pressure is obtained in the chambers A and B. Let w be the area of the plunger in square feet, p the difference of pressure in the chambers A and B in pounds per square foot, 'w the weight of the plunger and valve. Then if at any moment pm exceeds 'w the plunger will rise, and if it is less than the plunger will descend. Apart from friction, and assuming the valve D to be strictly an equilibrium valve, since cu and w are constant, p must be constant also, and equal to 'w/<.1>. By making w small and w large, the difference of pressure required to ensure the working of the apparatus may be made very small. Valves working with a difference of pressure of é in. of water have been constructed. VI. STEADY FLOW OF COMPRESSIBLE FLUIDS. § 61. External Work during the Expansion of Air.-If air expands without doing any external work, its temperature remains constant. This result was 'first

1 experimentally demonstrated

by ]. P. ]oule.

It leads to the conclusion

that, however air

changes its state, the internal

work done is pro:

portional to the change

5 6 of temperature. When,

in expanding, air does

work against an external

resistance, either heat

must be supplied or the

temperature falls.

To fix the conditions,

F - - suppose I lb of air confined behind a piston of

I sq. ft. area (fig. 76).

Let the initial pressure

be P1 and the volume of

the air 1/1, and suppose

this to expand to the

pressure pg and volume

v1. If p and v are the corresponding pressure and volume at any intermediate point in the expansion, the work done on the piston during the expansion from v to v-|-do is pda, ” and the whole work during the expansion from v1 to 112, represented by the area abcd, is 113° Pi /iii

V

< I

-... . .... ...... ., /gut... ...... ~, § A

»

FIG.A76. A K

v2

vlpdv.

Amongst possible cases two may be selected. Case x.-So much heat is supplied to the air during expansion that the temperature remains constant. Hyperbolic expansion. Then po = Pim-Work

done during expansion per pound of air = Zlz d 5 've

vlp 7) P1711 md?)/7)

-1171211 logs 'U2/711 ="'P1'U1l0g€ P1/pg. (I) Since the weight per cubic foot is the reciprocal of the volume per pound, this may be written

(P1/G1) logs G1/GQ. (IG)

Then the expansion curve ab is a common hyperbola. Case 2.-No heat is supplied to the air during expansion. Then the air loses an amount o heat equivalent to the external work done and the temperature falls. Adiabatie expansion. In this case it can be shown that P7/V = plvlys

Where 'y is the ratio of the specific heats of air at constant pressure and volume. Its value for air is I'408, and for dry steam 1-135. Work done during expansion per pound of air. = Y;ipda=p1o1~/ gida/oy

= -{pim/(~v-I)}{I/v2""- I/v1""}

=lpivn//(7-I)}{I/vt"“'-I/v2'“'}

=lP12/1/('Y-Ilill'(7'i/Wz)y'1}~ (2) The value of 121111 for any given temperature can be found from the data already given.

As before, substituting the weights G1, G2 per cubic foot for the volumes per pound, we get for the work of expansion (Pi/G1)lI/(V-I)} ll - (G2/G1)'”'l- (211) =1>1v1{I/(v ~ I)} II - (P2/P1)“"'/Yl- (2l>) § 62. Modification of the Theorem of Bernoulli for the Case of a Compressible Fluid.—In the application of the principle of work to a filament of compressible fluid, the internal work done by the expansion of the fluid, or absorbed

in its compression, must be A A

taken into account. Suppose, ~|"P'-ii B B as before, that AB (fig. 77) 'comes to A'B' in a short time t.

Let 121, w1, 111, G1 be the pressure, sectional area of stream,

velocity and weight of a cubic

foot at A, and pg, wi, 112, G2 the same quantities at B. Then, from the steadiness of motion, the weight of fluid passing A in any given time must be equal to the Weight passing B:

FIG. 77.

G1w1U1t = Ggwgvgf.

Let 21, 211 be the heights of the sections A and B above any given datum. Then the work of gravity on the mass AB in l seconds is G1w1v1l5(Z1 ~Z2) = W(21 -Z2)f»

where W is the weight of gas passing A or B per second. As in the case of an incompressible fluid, the work of the pressures on the ends of the mass AB is

Piwivii - P2w2'U2f1

=(P1/G1 -P2/G2)Wl.

The work done by expansion of Wt Ib of fluid between A and B is Wtffipdv. The change of kinetic energy as before is (W/eg) (1122-'u12)t. Hence, equating work to change of kinetic energy, we. -zr)¢+(1>1/G1-122/G2)WH-W2;jP<l°v= <w/2g> of-».2>¢; Z1-l°Pi/G1'l"iJ12/2g=Z2+P2/G2'l'v22/Zg' giipdv. (I) Now the work of expansion per pound of fluid has already been given. If the temperature is constant, we get (eq. Ia, § 61) Zi'l'Pi/G1+'Ui?/2g =-'52 +172/G2 +7122/23 ” (P1(Gl) losf (GI/G2)-But at constant temperature p1/G1 =p1/Gz; 21+v12/2§ =22+v22/2§ - (P1/Gi) loaf (P1/P2)1 (2) or, neglecting the difference of level, (1122 -'l/12)/2g =(P1/Gi) logs (P1/P2)- (25) Similarly, if the expansion is adiabatic (eq. 2a, § 61), Z1'i'p1/G1 +7112/2g = Z2+p2/G2 +7122/23 '(P1/G1)l1/(7 * lI~(1>2/Z>1)'I'”"/l; (3)

or neglecting the difference of level (1122 -1112)/2g = (Pl/G1)[I +I/(7 -° I)l1 (P2/P1)('(”'>/'/ll -P2/Gi (30) It will be seen hereafter that there is a limit in the ratio p1/p2 beyond which these expressions cease to be true. § 63. Discharge of Air from an Orifice.—The form of the equation of work for a steady stream of compressible fluid is 21 +P1/G1 +1/12/23 = Z2 +172/G2 +7122/25 °'(P1/G1)il/('Y ' I)l 1 ll -(ba/PO( WI, the expansion being adiabatic, because in the How of the streams of air through an orifice no sensible amount of heat can be communicated from outside.

Suppose the air Hows from a vessel, where the pressure is pr and the velocity sensibly zero, through an orifice, into a space where the pressure is pg. Let 112 be the velocity of the jet at a point where the convergence of the streams has ceased, -so that the pressure in the jet is also p2. As air is light, the work of gravity will be small compared with that of the pressures and expansion, so that 2122 may be neglected. Putting these values in the equation above-P1/G1 =P2/Girl-7/22/2§ (P1/Gill!/(7 - Ill ll '° (Pa/P1)(y' I)/7? vf/2g=1>1/G1-pr/Gz+(pi/Gull/lv-I)l li -<p./p.><*-'W = <z>1/Gm/(if - 1> - <z>./1>.>”- '/Y/<1 - nl -pl/Gi. But P1/G17 = P2/G2y P2/G2 = (P1/ Gi) (P2/ POW- 'W 7122/2g = fP1/G1)l'Y/('Y- I)} {I - (172/Pllwn I)/i/li (I) Of 1/22/28 = l'Y/l'Y“ Ill l(P1/Gi) " (Pi/Gzll; an equation commonly ascribed to L. J. Weisbach (Civilingenieur, 1856), though it appears to have been given earlier by A. [ C. Barre de Saint /enant and L Wantzel.

It has already (§ 9, eq. 4a) been seen that P1/G1 = (Po/Go) (Ti/fo)

where for air po =2I I6~8, G0 = -08075 and T0 =492-6. we/2g = lPof1“//Gofo(v- I)} {I - (PQ/P1)( ' 'l/ll? (2) or, inserting numerical values,

vi”/2g = 183'6f1{ I ~ (Pi/z>1)°'2°l; (20) which gives the velocity of discharge 212 in terms of the pressure and absolute temperature, pr, rl, in the vessel from which the air Hows, and the pressure pg in the vessel into which it Hows. Proceeding now as for liquids, and putting w for the area of the orifice and c for the coefficient of discharge, the volume of air discharged per secondat the pressure p2 and temperature 1-2 is Q2 =C<v2'2 =C°~' N/l(2£7P1/(7 - I)G1l (I - (P2/P1l(y I>/Yll = 103'7¢w V lfifl (P2/f>i)“'“"ll- (3)

If the volume discharged is measured at the pressure pr and absolute temperature 1-1 in the vessel from which the air flows, let Q1 be that volume; then

P1Q17 =P2Q2l:

Q1 = (P2/P1l'/lQ2¥

Q1=cw v Llzgvp./<v-1>G.}, i<z>2/1>1>'”- <1>2/1>1>'** 'Wil-Let (Pi/P02/Y (P2/f?1)( ' ' 9” 7 = (22/P1)1"“ " (112/Pill" =¢§ then Qi =cw / [2g'YP1¢/f'Y* I)G1l

= 108-7cw / (rn//). (4)

The weight of air at pressure pi and temperature -rl is G, =p1/53-21| lb per cubic foot.

Hence the weight of air discharged is

fV = GIQI =Cw N/ l2g'YPiG1/('Y' Ill

=2'043C°°P1 N/ ll///T1)~ (5)

Weisbach found the following values of the coefficient of discharge cz-

Conoidal mouthpieces of the form of the

contracted vein with effective ressures c-0-97 to o-99

Circular sharp-edged orifices .

Short cylindrical mouthpieces

The same rounded at the inner end

Conical converging mouthpieces

P

of -23 to 1-1 atmosphere .

0'563 ., 0-783

o-81, , 0'S4

0'92 H 0'93

0'90 H 0'99

§ 64. Limit to the Application of the above Formulae.-In the formulae above it is assumed that the fluid issuing from the orifice expands from the pressure pr to the pressure pz, while passing from the vessel to the section of the jet considered in estimating the area w. Hence pg is strictly the pressure in the jet at the plane of the external orifice in the case of mouthpieces, or at the plane of the contracted section in the case of simple orifices Till recently it was tacitly assumed that this pressure pg was identical with the general pressure external to the orifice. R. D. Napier first discovered that, when the ratio pg/pl exceeded a value which does not greatly differ from 0-5, this was no longer true. In that case the ex ansion of the fluid down to the external pressure is not completed pat the time it reaches the plane of the contracted section, and the pressure there is greater than the general external pressure; or, what amounts to the same thing, the section of the jet where the expansion is completed is a section which is greater than the area can of the contracted section of the jet, and may be greater than the area w of the orifice. Napier made experiments with steam which showed that, so long as pq/pi> 0-5, the formulae above were trustworthy, when pg was taken to be the general external pressure, but that, if p2/pi <0-5, then the pressure at the contracted section was independent of the external pressure and equal to 0-5p1. Hence in such cases the constant value o~5 should be substituted in the formulae for the ratio of the internal and external pressures pg/pl.

It is easily deduced from Weisbach's theory that, if the pressure external to an orifice is gradually diminished, the weight of air discharged per second increases to a maximum for a value of the ratio pl/or ={2/<v+1>}Yi'/Y

=o-527 for air

=o'58 for dry steam.

For a further decrease of external pressure the discharge diminishes, -a result no doubt improbable. The new view of Weisbach's formula is that from the point where the maximum is reached, or not greatly differing from it, the pressure at the contracted section ceases to diminish.

A. F. Fliegner showed (Civilingenieur xx., IS74) that for air Howing from well-rounded mouthpieces there is no discontinuity of the law of How, as Napier's hypothesis implies, but the curve of How bends so sharply that Napier's rule may be taken to be a good approximation to the true law. The limiting value of the ratio pg/pr, for which Weisbach's formula, as originally understood, ceases to apply, is for air o~5767; and this is the number to be substituted for pg/pl in the formulae when pg/pl falls below that value. For later researches on the How of air, reference may be made to G. A. Zeuner's pape§ (Cioilingenieur, 1871), and F1iegner's papers (ibid., 1877, 1878

VII. FRICTION OF LIQUIDS.

§ 65. When a stream of fluid flows over a solid surface, or conversely when a solid moves in still fluid, a resistance to the motion is generated, commonly termed fluid friction. It is due to the viscosity of the fluid, but generally the laws of fluid friction are very different from those of simple viscous resistance. It would appear that at all speeds, except the slowest, rotating eddies are formed by the roughness of the solid surface, or by abrupt changes of velocity distributed throughout the fluid; and the energy expended in producing these eddying motions is gradually lost in overcoming the viscosity of the fluid in regions more or less distant from that where they are first produced. »

The laws of fluid friction are generally stated thus:- I. The frictional resistance is independent of the pressure between the fluid and the solid against which it flows. This may be verified by a simple direct experiment. C. H. Coulomb, for instance, oscillated a disk under water, first with atmospheric pressure acting on the water surface, afterwards with the atmospheric pressure removed. No difference in the rate of decrease of the oscillations was observed. The chief proof that the friction is independent of the pressure is that no difference of resistance has been observed in water mains and in other cases, where water flows over solid surfaces under widely different pressures.

2. The frictional resistance of large surfaces is proportional to the area of the surface.

3. At low velocities of not more than 1 in. per second for water, the frictional resistance increases directly as the relative velocity of the fluid and the surface against which it Hows. At velocities of ~§ ft. per second and greater velocities, the frictional resistance is more nearly proportional to the square of the relative velocity. In many treatises on hydraulics it is stated that the frictional resistance is independent of the nature of the solid surface. The explanation of this was supposed to be that a film of fluid remained attached to the solid surface, the resistance being generated between this fluid layer and layers more distant from the surface. At extremely low velocities the solid surface does not seem to have much inHuence on the friction. In Coulomb's experiments a metal surface covered with tallow, and oscillated in water, had exactly the same resistance as a clean metal surface, and when sand was scattered over the tallow the resistance was only very slightly increased. The earlier calculations of the resistance of water at higher velocities in iron and wood pipes and earthen channels seemed to give a similar result. These, however, were erroneous, and it is now well understood that differences of roughness of the solid surface very greatly inHuence the friction, at such velocities as are common in engineering practice. H. P. G. Darcy's experiments, for instance, showed that in old and incrusted water mains the resistance was twice or sometimes thrice as great as in new and clean mains. § 66. Ordinary Expressions for Fluid Friction at Velocities not Extremely Small:-Let f be the frictional resistance estimated in pounds per square foot of surface at a velocity of 1 ft. per second; co the area of the surface in square feet; and 1; its velocity in feet per second relatively to the water in which it is immersed. Then, in accordance with the laws stated above, the total resistance of the surface is f

R =f.., ~U2 (1)

where f is a quantity approximately constant for any given surface. If

5 =2§ f/G,

R = i§ (j-o.>1/2/zg, (2)

where 5 is, like f, nearly constant for a given surface, and is termed the coefficient of friction.

The following are average values of the coefficient of friction for water, obtained from experiments on large plane surfaces, moved in an indefinitely large mass of water.

Frictional

Resistance in

lb perfsq. ft.

New well-painted iron plate . . . ~Oo489 °OO473 Painted and planed plank (Beaufoy) -00350 '00339 Surface of iron ships (Rankine) . -00362 ~0035I Varnished surface (Froude) . . -00258 -00250 Ifine sand surface, , . °004I8 °O0405 | Coarscr sand surface, ,, . . ~00503 -00488 The distance through which the frictional resistance is overcome is 1' ft. per second. The work expended in fluid friction is therefore given by the equation-Work

expended =ft.>v3 foot-pounds per second Q (3). =~5Gwv“/2g, . .. 5

The coefficient of friction and the friction per square foot of surface can be indirectly obtained from observations of the discharge of pipes and canals. In obtaining them, however, some assumptions as to the motion of the water must be made, and it will be better therefore to discuss these values in connexion with the cases to which they are related.

Many attempts have been made to express the coefficient of friction in a form applicable to low as well as high velocities. The older hydraulic writers considered the the resistance measured. For two planks differing in area by 46 sq. ft., at a velocity of 10 ft. per second, the difference of resistance, measured on the difference of area, was 0-33,9 lb per square foot. Also the resistance varied as the 1~949th power of the velocity. § 68. Froudelr Experiments:-The most important direct experiments on fluid friction at ordinary velocities are those made by William Froude (1810-1879) at Torquay. The method adopted in these experiments was to tow a board in a still water canal, the velocity and the resistance being registered by very ingenious recording arrangements. The general arrangement of the apparatus is shown in fig. 79. AA is the board the resistance of which is to be determined. B is a cut-water giving a fine entrance to the plane surfaces of the board. CC is a bar to which the board AA is attached, and which is suspended by a parallel motion from a carriage running on rails above the still water canal. G is a link by which the resistance of the board is transmitted to a spiral spring H. A bar I rigidly connects the other end of the spring to the carriage. The dotted lines K, L indicate the position of a couple of levers by which the extension of the spring is caused to move a pen M, which records the extension on a greatly increased scale, by a line drawn on the paper cylinder N. T his cylinder revolves at a speed proportionate to that of the carriage, its motion being obtained from the axle of the carriage Wheels. A second pen O, receiving jerks at every second and a quarter from a clock P, records time on the paper cylinder. The scale for the line of resistance is ascertained by stretching the spiral spring by known weights. The boards used for the experiment resistance termed fluid friction to be P 0 ', ,, made up of two parts, -a part due ' ", ';r:";;“;'; *' Y directly to the distortion of the mass of 5 water and proportional to the velocity '" " ' ' '§ F of the water relatively to the solid sur- |11 Lada, Rau N r t I face, and anotlaer pa1l;t due to kiplptic " A QI J /i energy imparte to the water stri in 5 ' ' ' the roughness es of the solid surface ang // 'E proportionfirl to the square of the ' I I I-, Z/ i..i': ve ocity. ence they proposed to take | l W if |?§ i ~., ,

E=a+5/U . IT' ' '"""" V

in which expression the second term is 1- 5 '=”' C / I C' -9 if * of greatest importance at very low H r velocities, and of comparatively little " "“"' " ' "'" ' " “ ' f " =" importance at velocities over about é ft.:':;“” ' ";4 °i };-1-I-;°-ifper second. Values of 5 expressed in this ” ”” " " "' °i T “ " " ".i; and similar forms will be given in con- i T;~ -; -; A A B # - °';- j' § nexion with' pipes and canals. ~ - - °. - — — — All these expressions must at present *I.:' ” V 1 " °l; 'f' ”' . f'.i 'f be regarded as merely empirical ex-""~"'—"”“”""' '” **““ ' -'“-***""'°' *"" press ions serving practical purposes. The frictional resistance will be seen to vary through wider limits than these expressions allow, and to depend on circumstances of which they do not take account.

§ 67. Coulomb's Experiments.-The first direct experiments on fluid friction were made by Coulomb, who employed a circular disk suspended by a thin brass wire and oscillated in its own plane. His experiments were chiefly made at very low velocities. When the disk is rotated to any given angle, it oscillates under the action of its inertia and the torsion of the wire. The oscillations diminish gradually in consequence of the work done in overcoming the friction pf the disk. The diminution furnishes a means of determining the riction.

Fig. 78 shows Coulomb's apparatus. LK supports the wire and disk; ag is the brass wire, the torsion of which causes the oscillations; DS is a graduated

1, K disk serving to measure

the angles through which

L the apparatus oscillates.

To this the friction disk

is rigidly attached hanging

in a vessel of water.

The friction disks were

9 S from 4~7 to 7~7 in. diap

meter, and they gener~

—f+~*"' ally made one oscillation

in from 20 to 30 seconds,

through angles varying

W from 3601 to 6°. When

}, ;.l, g "|l'll'l*{ the velocity of the cirll"" cumference of the disk

was less than 6 in. per

second, the resistance

was sensibly proportional

to the velocity.

Beauf03"s Ex;>e:'i1ne1zls.-Towards the end of the 18th century Colonel Mark Bcaufoy (1764-1827) made an immense mass of experiments on the rt-sistzuice of bodies moved through water (Nautical and Hyrlrazr/if ECA'[J€7'I'7Vl€1Zf5, London, 1834). Of these the only ones directly bearing on surface friction were some made in 1796 and 1798. Smooth painted planks were drawn through water and Fig. 78.

Fig. 79.- 1

were 135 in. thick, IQ in. deep, and from 1 to 50 ft. in length, cutwater included. A lead keel counteracted the buoyancy of the board. The boards were covered with various substances, such as paint, varnish, Hay's composition, tinfoil, &e., so as to try the effect of different degrees of roughness of surface. The results obtained by Froude may be summarized as follows:- 1. The friction per square foot of surface varies very greatly for different surfaces, being generally rcater as the sensible roughness of the surface is greater. Thus, wiien the surface of the board was covered as mentioned below, the resistance for boards 50 ft. long, at 10 ft. per second, was-Tinfoil

or varnish

Calico . .

Fine sand .

Coarser sand ......

0-25 Ib per sq. ft.

- 0'47 H .1

0-405 .. ..

0433 .. .»

2. The power of the velocity to which the friction is proportional varies for different surfaces. Thus, with short boards 2 ft. long, For tinfoil the resistance varied as vm". For other surfaces the resistance varied as v2'°°. With boards 50 ft. long,

For varnish or tinfoil the resistance varied as z»1'B3. For sand the resistance varied as rim". 3. The average resistance per square foot of surface was much greater for short than for long boards; or, what is the same thing; the resistance per square foot at the forward part of the board was greater than the friction per square foot of portions more stern ward. Thus,

Mean Resistance in

lb per sq. ft.

Varnished surface . 2 ft. long 0-41

50, , 0-25

Fine sand surface 2, , 0-81

50 H 0'405

This remarkable result is explained thus by Froude: “ The portion of surface that goes first in the line of motion, in experiencing resistance from the water, must in turn communicate motion to the water, in the direction in which it is itself travelling. Consequently the p0rtion of surface which succeeds the first will be rubbing, not against stationary water but against water partiall movin in its I I Y E

pwn direction, and cannot therefore experience so much resistance rom it.

§ 69. The following table gives a general statement of Froude's results. In all the experiments in this table, the boards had a fine cutwater and a fine stern end or run, so that the resistance was entirely due to the surface. The table gives the resistances per square foot in pounds, at the standard speed of 600 feet per minute, and the power of the speed to which the friction is proportional, so that the resistance at other speeds is easily calculated. the results obtained with the disks and Froude's results on olanks sponding to any speed N. From these the values off and n can be deduced, f being the friction per square foot at unit velocity. For comparison with Fr0ude's results it is convenient to calculate the resistance at 10 ft. per second, which is F=f10” The disks were rotated in chambers 22 in. diameter and 3, 6 and I2 in. deep. In all cases the friction of the disks increased a little as the chamber was made larger. This is probably due to the stilling of the eddies against the surface of the chamber and the feeding back of the stilled water to the disk. Ilence the friction depends not only on the surface of the disk but to some extent on the surface of the chamber in which it rotates. If the surface of the chamber is made rougher by covering with coarse sand there is also an increase of resistance.

For the smoother surfaces the friction varied as the 1~85th power of the velocity. For the rougher surfaces the power of the velocity to from I-9 to 2°I. This is in agreement with A V B C which the resistance was proportional varied Froude's results.

Experiments with a bright brass disk showed that the friction decreased with increase of 2 temperature. The diminution between 41° T06 4;4 4 3' and I;§ O° F. amounted to 18 % In the general 4 5 337 equation M =cN" for any given disk, I I Length of Surface, or Distance from Cutwater, in feet. W 2 ft. 8 ft. 20 ft. 50 ft.

4 A B c A B c I A B c

I Varnish 2-OO 390 325 264 278 240 I-83 -250 °226 Paraffin . 370 314 260 271 237

Tinloil 2 16 295 278 263 262 244 I-83] -246 °232 Calico I 93 I 725 626 504 531 447 I-87 — Fine sand 2 oo 690 583 450 480 3.84

Medium sand 2~OO 730 625 488 534 465 2-oo -488 -456 Coarse sand 2-00 I- 880 714 520 588 490 c¢=0-1328(I ~0-o02It),

where Q is the value of c for a bright brass Columns A give the power of the speed to which the resistance is approximately proportional.

Columns B give the mean resistance per square foot of the whole surface of a board of the lengths stated in the table. Columns C give the resistance in pounds of a square foot of surface at the distance stern ward from the cutwater stated in the heading. Although these experiments do not directly deal with surfaces of greater length than 50 ft., they indicate what would be the resistances of longer surfaces. For at 50 ft. the decrease of resistance for an increase of length is so small that it will make no very great difference in the estimate of the friction whether we suppose it to continue to diminish at the same rate or not to diminish at all. For a varnished surface the friction at IO ft. per second diminishes from 0~4.I to O'32 Th per square foot when the length is increased from 2 to 8 ft., but it only diminishes from 0~27S to O'25O lb per square foot for an increase from 20 ft. to 50 ft. I

If the decrease of friction stern wards is due to the generation of a current accompanying the moving plane, there is not at first sight any reason why the decrease should not be greater than that shown I by the experiments. The current accompanying the board might be assumed to gain in volume and velocity stern wards, till the velocity was nearly the same as that of the moving plane and the friction per square foot nearly zero. That this does not happen appears to be due to the mixing up of the current with the still water surrounding it. Part of the water in contact with the board at any point, and receiving energy of motion from it, passes afterwards to distant regions of still water, and portions of still water are fed in towards the board to take its place. In the forward part of the board more kinetic energy is given to the current than is diffused into surrounding space, and the current gains in velocity. At a greater distance back there is an approximate balance between the energy communicated to the water and that diffused. The velocity of the current accompanying the board becomes constant or nearly constant, and the friction per square foot is therefore nearly constant also. § 70. Friction of Rotating Dis/as.-A rotating disk is virtually a surface of unlimited extent and it is convenient for experiments on friction with different surfaces at different speeds. Experiments carried out by Professor W. C. Unwin (Proc. Inst. Civ. Eng. lxxx.) are useful both as illustrating the laws of fluid friction and as giving data for calculating the resistance of the disks of turbines and centrifugal umps. Disks of IO, 15 and 20 in. diameter fixed on a vertical shafi were rotated by a belt driven by an engine. They were enclosed in a cistern of water between parallel top and bottom fixed surfaces. The cistern was suspended by three Hne wires. The friction of the disk is equal to the tendency of the cistern to rotate, and this was measured by balancing the cistern by a fine silk cord passing over a pulley and carrying a scale pan in which weights could be placed. 1 If w is an element of area on the disk moving with the velocity v, the friction on this element is foe", where f and n are constant for any given kind of surface. Let a be the angular velocity of rotation, R the radius of the disk. Consider a ring of the surface between r and I-f-df. Its area is 21rrdr, its velocity ar and the friction of this ring is f21rrdfa"r". The moment of the friction about the axis of rotation is: >1ra'{fr"'“'1dr, and the total moment of friction for the two sides of the disk is

l

l

hi = 41ro."ff§ 1"'l'2(]r = l41ra"/fn -1-3) }fR"+“. If N is the number of revolutions per sec., 1,1={2n+?7rn+1Nn/(n }j"R1|+3

and the work expended In rotating the disk is Mu. = {2"*'1r"+”N"+'/(n+3)}fR"+' foot lb per sec. The experiments give directly the values of M for the disks c0rredisk 0-85 ft. in diameter at a temperature t° F. The disks used were either polished or made rougher by varnish or by varnish and sand. The following table gives a comparison of 50 ft. long. The values given are the resistances per square foot at I0 ft. per sec.

Froudelv Experiments. Disk Experiments. Tinfoil surface O°232 Bright brass 0~2o2 to 0~229 Varnish ., .. 0-226 Varnish 0-220 to O'233 Fine sand . 0-337 Fine sand 0-339

Medium sand 0~456 Very coarse sand 0~587 to O'7I5 VIII. STEADY FLOW OF WATER IN PIPES OF UNIFORM SECTION.

§ 71. The ordinary theory of the flow of water in pipes, on which all practical formulae are based, assumes that the variation of velocity at different points of any cross section may be neglected. The water is considered as moving in plane layers, which are driven through the pipe against the frictional resistance, by the difference of pressure at or elevation of the ends of the pipe. If the motion is steady the velocity at each cross section remains the same from moment to moment, and if the cross sectional area is constant the velocity at all sections must be the same. Hence the motion is uniform. The most important resistance to the motion of the water is the surface friction of the pipe, and it is convenient to estimate this independently of some smaller resistances which will be accounted for presently.

In any portion of a uniform pipe, excluding for the present the ends of the pipe, the water enters and leaves at the same velocity. For that portion therefore

the work of the

external forces and of

the surface friction

must be equal. Let

fig. 80 represent a very

short portion of the

pipe, of length Ill, between

cross sections at

z and z-I-dz ft. above

any horizontal datum

line xx, the pressures at

the cross sections being

PJ?

p and p-I-dp lb per '?"square

foot. Further,

let Q be the volume of

$

2

-1l[ V> 5

""»-f "ff

1

"'- I


» A /PHI

T"'

é ', ;fl'-S

524112 9

4 —.LY

FIG. 80.

flow or discharge of the pipe per second, SZ the area of a normal cross section, and X the perimeter of the pipe. The Q cubic feet, which How through the space considered per second, weigh GQ lb, and fall through a height-dz ft. The work done by gravity is then -GQdz; »

a positive quantity if dz is negative, and vice versa. The resultant pressure parallel to the axis of the pipe is P—(p-l»dp) = -dp lb per square foot of the cross section. The work of this pressure on the volume Q is

-°QfiP-

The only remaining force doing work on the system is the friction against the surface of the pipe. The area of that surface is X dl. The work expended in overcoming the frictional resistance per second is (see § 66, eq. 3)

-§ GXdlv“/2g,

or, since Q==f2v, V

  • § 'G(X/S?)Q(1'”, /'2§ >1U;

the negative sign being taken because the work is done against a resistance. Adding all these portions of work, and equating the result to zero, since the motion is uniform,—

−GQ dz − Q dpζG (χ/Ω) Q (v2/2g) dl = 0.

Dividing by GQ,

dz + dp/G + ζ (χ/Ω) (v2/2g) dl = 0.

Integrating,

z + p/G + ζ (χ/Ω) (v2/2g) l = constant.
(1)

§ 72. Let A and B (fig. 81) be any two sections of the pipe for which p, z, l have the values p1, z1, l1, and p2, z2, l2, respectively. Then

z1 + p1/G + ζ (χ/Ω) (v2/2g) l1 = z2 + p2/G + ζ (χ/Ω) (v2/2g) l2;

or, if l2l1 = L, rearranging the terms,

ζv2/2g = (1/L) {(z1 + p1/G) − (z2 + p2/G)} Ω/χ.
(2)
Fig. 81.

Suppose pressure columns introduced at A and B. The water will rise in those columns to the heights p1/G and p2/G due to the pressures p1 and p2 at A and B. Hence (z1 + p1/G) − (z2 + p2/G) is the quantity represented in the figure by DE, the fall of level of the pressure columns, or virtual fall of the pipe. If there were no friction in the pipe, then by Bernoulli’s equation there would be no fall of level of the pressure columns, the velocity being the same at A and B. Hence DE or h is the head lost in friction in the distance AB. The quantity DE/AB = h/L is termed the virtual slope of the pipe or virtual fall per foot of length. It is sometimes termed very conveniently the relative fall. It will be denoted by the symbol i.

The quantity Ω/χ which appears in many hydraulic equations is called the hydraulic mean radius of the pipe. It will be denoted by m.

Introducing these values,

ζv2/2g = mh/L = mi.
(3)

For pipes of circular section, and diameter d,

m = Ω/χ = 1/4πd2/πd = 1/4d.

Then

ζv2/2g = 1/4dh/L = 1/4 di;
(4)

or

h = ζ (4L/d) (v2/2g);
(4a)

which shows that the head lost in friction is proportional to the head due to the velocity, and is found by multiplying that head by the coefficient 4ζL/d. It is assumed above that the atmospheric pressure at C and D is the same, and this is usually nearly the case. But if C and D are at greatly different levels the excess of barometric pressure at C, in feet of water, must be added to p2/G.

§ 73. Hydraulic Gradient or Line of Virtual Slope.—Join CD. Since the head lost in friction is proportional to L, any intermediate pressure column between A and B will have its free surface on the line CD, and the vertical distance between CD and the pipe at any point measures the pressure, exclusive of atmospheric pressure, in the pipe at that point. If the pipe were laid along the line CD instead of AB, the water would flow at the same velocity by gravity without any change of pressure from section to section. Hence CD is termed the virtual slope or hydraulic gradient of the pipe. It is the line of free surface level for each point of the pipe.

If an ordinary pipe, connecting reservoirs open to the air, rises at any joint above the line of virtual slope, the pressure at that point is less than the atmospheric pressure transmitted through the pipe. At such a point there is a liability that air may be disengaged from the water, and the flow stopped or impeded by the accumulation of air. If the pipe rises more than 34 ft. above the line of virtual slope, the pressure is negative. But as this is impossible, the continuity of the flow will be broken.

If the pipe is not straight, the line of virtual slope becomes a curved line, but since in actual pipes the vertical alterations of level are generally small, compared with the length of the pipe, distances measured along the pipe are sensibly proportional to distances measured along the horizontal projection of the pipe. Hence the line of hydraulic gradient may be taken to be a straight line without error of practical importance.

Fig. 82.

§ 74. Case of a Uniform Pipe connecting two Reservoirs, when all the Resistances are taken into account.—Let h (fig. 82) be the difference of level of the reservoirs, and v the velocity, in a pipe of length L and diameter d. The whole work done per second is virtually the removal of Q cub. ft. of water from the surface of the upper reservoir to the surface of the lower reservoir, that is GQh foot-pounds. This is expended in three ways. (1) The head v2/2g, corresponding to an expenditure of GQv2/2g foot-pounds of work, is employed in giving energy of motion to the water. This is ultimately wasted in eddying motions in the lower reservoir. (2) A portion of head, which experience shows may be expressed in the form ζ0v2/2g, corresponding to an expenditure of GQζ0v2/2g foot-pounds of work, is employed in overcoming the resistance at the entrance to the pipe. (3) As already shown the head expended in overcoming the surface friction of the pipe is ζ(4L/d) (v2/2g) corresponding to GQζ (4L/d) (v2/2g) foot-pounds of work. Hence

GQh = GQv2/2g + GQζ0v2/2g + GQζ·4L·v2/d·2g;

h = (1 + ζ0 + ζ·4L/d) v2/2g. v = 8.025 √ [hd / {(1 + ζ0)d + 4ζL} ].

(5)

If the pipe is bell-mouthed, ζ0 is about = .08. If the entrance to the pipe is cylindrical, ζ0 = 0.505. Hence 1 + ζ0 = 1.08 to 1.505. In general this is so small compared with ζ4L/d that, for practical calculations, it may be neglected; that is, the losses of head other than the loss in surface friction are left out of the reckoning. It is only in short pipes and at high velocities that it is necessary to take account of the first two terms in the bracket, as well as the third. For instance, in pipes for the supply of turbines, v is usually limited to 2 ft. per second, and the pipe is bellmouthed. Then 1.08v2/2g = 0.067 ft. In pipes for towns’ supply v may range from 2 to 41/2 ft. per second, and then 1.5v2/2g = 0.1 to 0.5 ft. In either case this amount of head is small compared with the whole virtual fall in the cases which most commonly occur.

When d and v or d and h are given, the equations above are solved quite simply. When v and h are given and d is required, it is better to proceed by approximation. Find an approximate value of d by assuming a probable value for ζ as mentioned below. Then from that value of d find a corrected value for ζ and repeat the calculation.

The equation above may be put in the form

h = (4ζ/d) [{ (1 + ζ0) d/4ζ} + L] v2/2g;
(6)

from which it is clear that the head expended at the mouthpiece is equivalent to that of a length

(1 + ζ0) d/4ζ

of the pipe. Putting 1 + ζ0 = 1.505 and ζ = 0.01, the length of pipe equivalent to the mouthpiece is 37.6d nearly. This may be added to the actual length of the pipe to allow for mouthpiece resistance in approximate calculations.

§ 75. Coefficient of Friction for Pipes discharging Water.—From the average of a large number of experiments, the value of ζ for ordinary iron pipes is

ζ = 0.007567.
(7)

But practical experience shows that no single value can be taken applicable to very different cases. The earlier hydraulicians occupied themselves chiefly with the dependence of ζ on the velocity. Having regard to the difference of the law of resistance at very low and at ordinary velocities, they assumed that ζ might be expressed in the form

ζ = a + β/v.

The following are the best numerical values obtained for ζ so expressed:—

  α β
R. de Prony (from 51 experiments) 0.006836 0.001116
J. F. d’Aubuisson de Voisins 0.00673 0.001211
J. A. Eytelwein 0.005493 0.00143

Weisbach proposed the formula

4ζ = α + β/√v = 0.003598 + 0.004289/√v.
(8)

§ 76. Darcy’s Experiments on Friction in Pipes.—All previous experiments on the resistance of pipes were superseded by the remarkable researches carried out by H. P. G. Darcy (1803–1858), the Inspector-General of the Paris water works. His experiments were carried out on a scale, under a variation of conditions, and with a degree of accuracy which leaves little to be desired, and the results obtained are of very great practical importance. These results may be stated thus:—

1. For new and clean pipes the friction varies considerably with the nature and polish of the surface of the pipe. For clean cast iron it is about 11/2 times as great as for cast iron covered with pitch.

2. The nature of the surface has less influence when the pipes are old and incrusted with deposits, due to the action of the water. Thus old and incrusted pipes give twice as great a frictional resistance as new and clean pipes. Darcy’s coefficients were chiefly determined from experiments on new pipes. He doubles these coefficients for old and incrusted pipes, in accordance with the results of a very limited number of experiments on pipes containing incrustations and deposits.

3. The coefficient of friction may be expressed in the form ζ = α + β/v; but in pipes which have been some time in use it is sufficiently accurate to take ζ = α1 simply, where α1 depends on the diameter of the pipe alone, but α and β on the other hand depend both on the diameter of the pipe and the nature of its surface. The following are the values of the constants.

For pipes which have been some time in use, neglecting the term depending on the velocity;

ζ = α (1 + β/d).
(9)
  α β
For drawn wrought-iron or smooth cast-iron pipes .004973 .084
For pipes altered by light incrustations .00996 .084

These coefficients may be put in the following very simple form, without sensibly altering their value:—

For clean pipes ζ = .005 (1 + 1/12d)
For slightly incrusted pipes ζ = .01 (1 + 1/12d)
(9a)

Darcy’s Value of the Coefficient of Friction ζ for Velocities not less
than 4 in. per second.

Diameter
of Pipe
in Inches.
ζ Diameter
of Pipe
in Inches.
ζ
New
Pipes.
Incrusted
Pipes.
New
Pipes.
Incrusted
Pipes.
2  0.00750 0.01500 18 .00528 .01056
3 .00667 .01333 21 .00524 .01048
4 .00625 .01250 24 .00521 .01042
5 .00600 .01200 27 .00519 .01037
6 .00583 .01167 30 .00517 .01033
7 .00571 .01143 36 .00514 .01028
8 .00563 .01125 42 .00512 .01024
9 .00556 .01111 48 .00510 .01021
12.00542 .01083 54 .00509 .01019
15 .00533 .01067      

These values of ζ are, however, not exact for widely differing velocities. To embrace all cases Darcy proposed the expression

ζ = (α + α1/d) + (β + β1/d2) /v;
(10)

which is a modification of Coulomb’s, including terms expressing the influence of the diameter and of the velocity. For clean pipes Darcy found these values

α= .004346
α1= .0003992
β= .0010182
β1= .000005205.

It has become not uncommon to calculate the discharge of pipes by the formula of E. Ganguillet and W. R. Kutter, which will be discussed under the head of channels. For the value of c in the relation v = c √(mi), Ganguillet and Kutter take

c = 41.6 + 1.811/n + .00281/i
1 + [ (41.6 + .00281/i) (n/ √m) ]

where n is a coefficient depending only on the roughness of the pipe. For pipes uncoated as ordinarily laid n = 0.013. The formula is very cumbrous, its form is not rationally justifiable and it is not at all clear that it gives more accurate values of the discharge than simpler formulae.

§ 77. Later Investigations on Flow in Pipes.—The foregoing statement gives the theory of flow in pipes so far as it can be put in a simple rational form. But the conditions of flow are really more complicated than can be expressed in any rational form. Taking even selected experiments the values of the empirical coefficient ζ range from 0.16 to 0.0028 in different cases. Hence means of discriminating the probable value of ζ are necessary in using the equations for practical purposes. To a certain extent the knowledge that ζ decreases with the size of the pipe and increases very much with the roughness of its surface is a guide, and Darcy’s method of dealing with these causes of variation is very helpful. But a further difficulty arises from the discordance of the results of different experiments. For instance F. P. Stearns and J. M. Gale both experimented on clean asphalted cast-iron pipes, 4 ft. in diameter. According to one set of gaugings ζ = .0051, and according to the other ζ = .0031. It is impossible in such cases not to suspect some error in the observations or some difference in the condition of the pipes not noticed by the observers.

It is not likely that any formula can be found which will give exactly the discharge of any given pipe. For one of the chief factors in any such formula must express the exact roughness of the pipe surface, and there is no scientific measure of roughness. The most that can be done is to limit the choice of the coefficient for a pipe within certain comparatively narrow limits. The experiments on fluid friction show that the power of the velocity to which the resistance is proportional is not exactly the square. Also in determining the form of his equation for ζ Darcy used only eight out of his seventeen series of experiments, and there is reason to think that some of these were exceptional. Barré de Saint-Venant was the first to propose a formula with two constants,

dh/4l = mVn,

where m and n are experimental constants. If this is written in the form

log m + n log v = log (dh/4l),

we have, as Saint-Venant pointed out, the equation to a straight line, of which m is the ordinate at the origin and n the ratio of the slope. If a series of experimental values are plotted logarithmically the determination of the constants is reduced to finding the straight line which most nearly passes through the plotted points. Saint-Venant found for n the value of 1.71. In a memoir on the influence of temperature on the movement of water in pipes (Berlin, 1854) by G. H. L. Hagen (1797–1884) another modification of the Saint-Venant formula was given. This is h/l = mvn/dx, which involves three experimental coefficients. Hagen found n = 1.75; x = 1.25; and m was then nearly independent of variations of v and d. But the range of cases examined was small. In a remarkable paper in the Trans. Roy. Soc., 1883, Professor Osborne Reynolds made much clearer the change from regular stream line motion at low velocities to the eddying motion, which occurs in almost all the cases with which the engineer has to deal. Partly by reasoning, partly by induction from the form of logarithmically plotted curves of experimental results, he arrived at the general equation h/l = c (vn/d3−n) P2−n, where n = l for low velocities and n = 1.7 to 2 for ordinary velocities. P is a function of the temperature. Neglecting variations of temperature Reynold’s formula is identical with Hagen’s if x = 3−n. For practical purposes Hagen’s form is the more convenient.

Values of Index of Velocity.

Surface of Pipe. Authority. Diameter
of Pipe
 in Metres. 
Values of n.
Tin plate Bossut .036 1.697 1.72
.054 1.730
Wrought iron (gas pipe)  Hamilton Smith  .0159 1.756 1.75
.0267 1.770
Lead Darcy .014 1.866 1.77
.027 1.755
.041 1.760
Clean brass Mair .036 1.795   1.795 
Asphalted Hamilton Smith .0266 1.760 1.85
Lampe. .4185 1.850
W. W. Bonn .306 1.582
Stearns 1.219 1.880
Riveted wrought iron Hamilton Smith .2776 1.804 1.87
.3219 1.892
.3749 1.852
Wrought iron (gas pipe) Darcy .0122 1.900 1.87
.0266 1.899
.0395 1.838
New cast iron Darcy .0819 1.950 1.95
.137 1.923
.188 1.957
.50 1.950
Cleaned cast iron Darcy .0364 1.835 2.00 
.0801 2.000
.2447 2.000
.397 2.07
Incrusted cast iron Darcy .0359 1.980 2.00
.0795 1.990
.2432 1.990

FIG. 83.

In 1886, Professor W. C. Unwin plotted logarithmically all the most trustworthy experiments on flow in pipes then available? Fig. 83 gives one such plotting. The results of measuring the slopes of the lines drawn through the plotted points are given in the table.

It will be seen that the values of the index n range from I-72 for the smoothest and cleanest surface, to 2-00 for the roughest. The numbers after the brackets are rounded off numbers. The value of n having been thus determined, values of m/df were next found and averaged for each pipe. These were again plotted logarithmically in order to find a value for x. The lines were not very regular, but in all cases the slope was greater than I to I, so that the value of x must be greater than unity. The following table gives the results and a comparison of the value of x and Reyn0lds's value 3~:1..

Kind of Pipe. n 3-71 x

Tin plate, ... x~72 1-28 1-ioo

Wrought iron (Smith). 1-75 1-25 1-210 Asphaltcd pipes 1-85 I-15 I-127 lWrought iron (Darcy) 1-87 I-13 1-680 1 Riveted wrought iron 1-87 I°I3 1-390 1New cast iron I-95 I'O5 I-168 Cleaned cast iron 2~00 1-00 I-168 1lncrustecl cast iron 2-00 I-00 I~I6O With the exception of the anomalous values for Darcy's wrought iron pipes, there is no great discrepancy between the values of x and 3-n, but there is no appearance of relation in the two quantities. Ftor the present it appears preferable to assume that x is independent 0 n.

It is now possible to obtain lL s f tl h' d 1 IC 0 ie t ir constant m, using the values found for 71 and The following table gives the results, the values of m being for metric measures. Tffformulae for the Flow of 'ate1TmTPipes, ” Industries i(Manchester, 1886).

1

I

Here, considering the great range of diameters and velocities in the experiments, the constancy of m is very satisfactorily close. The asphalted pipes give iather variable values. But, as some of these were new and some old, the variation is, perhaps, not surprising. The incrusted pipes give a value of m quite double that for new pipes but that is perfectly consistent with what is known of fluid friction in other cases.

I

Diameter V 1 e f'1T~Mean

Kind of Pipe. Min a:Z O Value Authority. etres. ' 0 m.

Tin plate . gzggi '01686 Bossut Wrought iron gig; 'gigii -01310 Hamilton Smith f0~027 'O174Q1 Hamilton Smith

0-306 02058 W. W. Bonn

Asphalted 0-306 -02107 ' MSM W. W. Bonn pipes 0-419 01650 Lampe

I'2I9 ~0I3I7 Stearns

-219 ~O2107. Gale

1'0-278 ~013701

- 0-322 01440

Rweted - < 0-375 01390 s -01403 Hamilton Smith wrought iron 0432 01368

-0-657 ~O144S:l

0°O82 -01725 1

New cast iron < 1* '01658 Dafcb ><>-500 -017459 1

C1 d 0'08O (31979

fig? Cagt < 0-245 -02091 1' -01994 1 Darcy >O~297 ~OI9I3£ 1

11O'()36 03693 1 1

Ingglxfted cast < 0-080 -03530 > -03643 Darcy 1 1 O-243 ~03Zo6 1 1 General Mean Values of Constants.

The general formula (Hagen's)-h/I=rno"/d'.2g-can therefore be taken to fit the results with convenient closeness, if the following mean values of the coefficients are taken, the unit being a metre:- This shows a marked decrease of resistance as the temperature rises. If Professor Osborne Reynolds's equation is assumed h=rnLV"/d'*“", and n is taken I-795, then values of rn at each temperature are practically constant-

Kind of Pipe. in x n

Wrought iron, . -0131 -21 1-75

Asphalted iron . . -0183 1-12 1-85

Riveted wrought iron

New cast iron

Cleaned cast iron

Incrusted cast iron

~OI4O

~0166

0199

I

I

I

I

7

390

-168

-168

1-87

I-95

2'O

Tin plate .... ~0I6Q I°I0 1-72

1

-0364

-160

The variation of each of these coefficients is within a comparatively narrow range, and the selection of the proper coefficient for any given case presents no difficulty, if the character of the surface of the pipe is known.

It only remains to give the values of these coefficients when the quantities are expressed in English feet. For English measures the following are the values of the coefficients:- Kind of Pipe. rn x n 1

Tin plate . . 0265 1-10 1-72

Wrought iron . . 0226 I~2I 1~75

Asphalted iron . 0254 1-127 1-85 Riveted wrought iron . 0260 1-390 1-87 New cast iron . 0215 1-168 1-95

Cleaned cast iron 0243 1~168 2'0 Incrusted cast iron . -0440 I'160 2-0 § 78. bistribiztion of Velocity -in the Cross Section of a Pipe.—Darcy made experiments with a Pitot tube in ISSO on the velocity at different points in the cross section of a pipe. He deduced the relation

V-v = 11~3(r%/R), /i,

where V is the velocity at the centre and v the velocity at radius r in a pipe of radius R with a hydraulic gradient i. Later Bazin repeated the experiments and extended tl1CII1 (/llém. de VA cadérnie des Sciences, xxxii. No. 6). The most important result was the ratio of mean to central velocity. Let b= Ri/UZ, where U is the mean velocity in the pipe; then V/U =1+9-o3, / b. A very useful result for practical purposes is that at 0-74 of the radius of the pipe the velocity is equal to the mean velocity. Fig. 84 gives the velocities at different radii as determined by Bazin.

§ 79. Influence of Temperature on the Flow through Pipes.-Very careful experiments on the flow through a pipe 0-1236 ft. in diameter Temp. F. rn. Temp. F. rn.

57 0-000276 100 O~O0O244

70 0000263 1 IO O'000235

80 0-0002 57 120 0~ooo229

90 O'OOO250 130 0~0OO225

160 0-000206

where again a regular decrease of the coefficient occurs as the temperature rises. In experiments on the friction of disks at different temperatures Professor W. C. Unwin found th-at the resistance was proportional to constant >< (1-0-o021t) and the values of rn given above are expressed almost exactly by the relation rn =o-o00311(1~o-00215 I).

In tank experiments on ship models for small ordinary variations of temperature, it is usual to allow a decrease of 3 22, of resistance for 10° F. increase of temperature.

§ 80. Influence of Deposits in Pipes on the Discharge. Scraping Water Mains.-The influence of the condition of the surface of a pipe on the friction is shown by various facts known to the engineers of Waterworks. In pipes which convey certain kinds of waterjoxidation proceeds rapidly and the discharge is considerably diminished. A main laid at Torquay in 1858, 14 m. in length, consists of 10~in., 9-in. and 8-in. pipes. It was not protected from corrosion by any coating. But it was found to the surprise of the engineer that in eight years the discharge had diminished to SI % of the original discharge. ]. G. Appold suggested an apparatus for scraping the interior of the pipe, and this was constructed and used under the direction of William Froude (see “ Incrustation of Iro11 Pipes, ” by W. Ingham, Proc. Inst. Mech. Eng., 1899). It was found that by scraping the interior of the pipe the discharge was increased 56 %. The scraping requires to be repeated at intervals. After each scraping the discharge diminishes rather rapidly to 10%) and afterwards more slowly, the diminution in a year being about 25 %. Fig. 85 shows a scraper for water mains, similar to Appold's but modified in details, as constructed by the Glenfield Company, at Kilmarnock. A is a longitudinal section of the pipe, showing the scraper in place; B is an end view of the plungers, and C, D sections of the boxes placed at intervals on the main for introducing or withdrawing the scraper. The apparatus consists of two plungers, packed with leather so as to fit the main pretty closely. On the spindle of these plungers are fixed eight steel scraping blades, with curved scraping edges fitting the surface of the main. The apparatus is placed in the main by removing the cover from one of the boxes shown at C, D. The cover is then replaced, water pressure is admitted behind the plungers, and the apparatus driven through the °°B . Les.:»;. :;: t-i., , s; swerves“~w~swwm®wa=<;a=::s===~; V333 ' 1 . -2,

I1 ~- 't 'r -.

5| .....:pigs | - r;7§ :>.::A :::, , , J ¢§ r, ==== :Qi ~ - - -;-was 1 °—"'- ld Y -1- ~~, Eg . ;t: .-::::;::::;:::q - a =- .

5, ...s af, -

  • FMS - JL- T

| | 3.5¢ / ,

L = »»mx<w m§ m@mw ws . '§ >m® Nes-w~“W~~xs-sx» ~; sxwwwwwssw l|lllHfEli|' f s»'§§§ i$i§ “* 7 § o ° o A V I

<§ B sa! "' "' ~§ ' "' " sq; 4 - an f .

MM - wma =-2 r-~ D

er s C r 2 I

1.145 ' r ° ° ° ' lr é 3

§ ° o, ':&;:;:: . ~ -~~ ° - rr, ln-me § i ao- o. /¢§ .az /

~ o sP my

|:)g;7 Q&Q&}§§ '$;®' . /, w g /../// mfr” 1///f, , fav / / 073

FIG. 84. Fm. 85. Scale 215.

and 25 ft. long, with water at different temperatures, have been made by J. G. Mair (Proc. Inst. Civ. Eng. lxxxiv.). The loss of head was measured from a point 1 ft. from the inlet, so that the loss at entry was eliminated. The 1% in. pipe was made smooth inside and to gauge, by drawing a mandril through it. Plotting the results logarithmically, it was found that the resistance for all temperatures varied very exactly as v“795, the index being less than 2 as in other experiments with very smooth surfaces. Taking the ordinary equation of How /t= § '(4I¢/D)§ v2/2g), then for heads varying from 1 ft. to nearly 4 ft., and velocities

second, the values of g' were

ln the pipe varying from 4 ft. to 9 ft. per as follows:-

Temp. F. g* Temp. F. I

57 -0044 to -0052 loo ~oo39 to -0042 70 ~oo4z to -0045 110 0037 to 0041 80 -0041 to -0045 120 -0037 to -0041 go ~oo4o to -0045 130 -0035 to -0039 160 -0035 to -0038

main. At Lancaster after twice scraping the discharge was increased 56% %, at Oswestry 54% %. The increased discharge is due to the diminution of the friction of the pipe by removing the roughness es due to oxidation. The scraper can be easily followed when the mains are about 3 ft. deep by the noise it makes. The average speed of the scraper at Torquay is 2% m. per hour. At Torquay 49% of the deposit is iron rust, the rest being silica, lime and organic matter. In the opinion of some engineers it is inadvisable to use the scraper. The incrustation is only temporarily removed, and if the use of the scraper is continued the life of the pipe is reduced. The only treatment effective in, preventing or retarding the incrustation due to corrosion is to coat the pipes when hot with a smooth and perfect layer of pitch. With certain waters such as those derived from the chalk the incrustation is of a different character, consisting of nearly pure calcium carbonate. A deposit of another character which has led to trouble in some mains is a black slime containing a good deal of iron not derived from the pipes. It appears to be an organic growth. Filtration of the water appears to prevent the growth of the slime, and its temporary removal may be effected by a kind of brush scraper devised by G. F. Deacon (see “ Deposits in Pipes, ” by Professor ]. C. Campbell Brown, Proc. Inst. Civ. Eng., 1903-1904).

§ 81. Flow of Water through Fire Hose.-The hose pipes used for fire purposes are of very varied character, and the roughness of the surface varies. Very careful experiments have been made by J. R. Freeman' (Arn. Soc. Civ. Eng. xxi., 1889). It was noted that under pressure the diameter of the hose increased sufficiently to have a marked influence on the discharge. In reducing the results the true diameter has been taken. Let v=mean velocity in ft. per sec.; r=hydraulic mean radius or one-fourth the diameter in feet; hydraulic gradient. Then t'=n/ (ri).

relative velocity of the two parts Diangleter @31g§ ;1§ 1

tates 1 1) 71.

Inches. per min 4

°

Solid rubber 5 2-65 215 -1863 12-50 123-3 hose .» 344 -4714 20°OO' 124-0

Woven cotton, 2-47 200 2464 I3-40' 119-1 rubber lined i, , 299 5269 2O°O0 121-5 Woven cotton, 2'49 200 2427 13-20 117-7 rubber lined i, , 319 5708 21-oo 122-1 Knit cotton, g 2-68 132 0809 7-50 111-6 rubber lined 2, , 299 3931 I7-O0 114-8 |' Knit cotton, 5 2-69 204 2357 11-50 100-1 3 rubber lined l, , 319 5165 I8~OO l05~8 Woven cotton, < 2-12 154 3448 I 4°0O 113-4 rubber lined 2, , 240 7673 21-81 118-4 Woven cotton, 2-53 54-8 0261 3-50 94-3 rubber lined i, , 298 8264 I9-OO 9I~O Unlined linen { 2-60 57-9 0414 3-50 73-9 hose, , 4 331 1624 '20°O0 79-6 § 82. Reduction of a Long Pipe of Varying Diameter to an Equivalent Pipe of Uniform Diameter. Dupuit's Equation.-Water mains for the supply of towns often consist of a series of lengths, the diameter being the same for each length, but differing from length to length. In approximate calculations of the head lost in such mains, it is generally accurate enough to neglect the smaller losses of head and to have regard to the pipe friction only, and then the calculations may be facilitated by reducing the main to a main of uniform diameter, in which there would be the same loss of head. Such a uniform main will be termed an equivalent main. - -:

f<- —» -[1 —~ ->i~= —»»-~ lg ~—~ ~—1-l<~ —- lg- -~—~ as u . I

- .

A 2:11 tl; I:ALI 'Y;

1 ¢

<-~~- - —~ Z- -.- . -»—e-91

B if .1

FIG. 86.

In fig. 86 let A be the main of variable diameter, and B the equivalent uniform main. In the given main of variable diameter A, let l, , 12... be the lengths,

di, dz... the diameters,

vi, vz... the velocities,

il, iQ... the slopes,

for the successive portions, and let l, d, v and i be corresponding quantities for the equivalent uniform main B. The total loss of head in A due to friction is

It = 1111 . . .

= I(U12'4l1/2£d1) +s“(f12”'4l2/23112) + ~ -and in the uniform main

il = tt-2-41/21-1>.

If the mains are equivalent, as defined above, s“(v”°4l/2gd) = i“(vF~4J1/21111)+s“(v2”~4li/2gd2)+ . .-But, since the discharge is the same for all portions, § 1rd'2v = %1rd127J1 = 341rd2'27J2 = . . . 171 = Udg/diz; v2=vd'/df . . .

Ziifo suppose that of may be treated as constant for all the pipes. ell

I/fi = (df/dif) (ll/dl) 'l' Q14/1124) (12/112) +—l- <d°/li1'°)l1+<d5/(l2°)l2+...

which gives the length of the equivalent uniform main which would have the same total loss of head for any given discharge. § 83. Other Losses of Head in Pipesf-Most of the losses of head in pipes, other than that due to surface friction against the pipe, are due to abrupt changes in the velocity of the stream producing eddies. The kinetic energy of these is deducted from the general energy of translation, and practically wasted. Sudden Enlargement of Section.-Suppose a pipe enlarges in section from an area wo to an area wi (fig. 87); then A

111/1/o==wo/wi? E

or, if the section is circular,

W1/'Un ='= (do/'diy:

The head lost at the abrupt change tof velocity has already been

shown to be the head due to the

  • v d

tl,

of the stream. Hence head lost FIG' 87 fit = (Uv -1102/22 = (w1/ wo- IW?/23 = {(fl1/do)”- 1}2”12/28 of f)¢=i'e7]12/Zgr (I)

if fe is put for the expression in brackets. w1/w0- 1.1 1.2 1.5 1 7 1.8 1.0 2.0 2.5 3.0 3.5 4.0 5.0 6.0 7.0 8.0 lil/d0= 1.05 1.10 1.22 1.30 1.34 1.38 1.41 1.58 1.73 1.87 2.00 2.24 2.45 2.65 2.83 g-e= .01 .04 .25 .49 .64 .81 1.00 2.25 4.00 6.25 9.oo16 0025.:>o 36.o49.o Abrupt Contraction of Section.~When water passes from a larger to a smaller section, as in hgs. 88, 89, a contraction is formed, and the contracted stream abruptly expands to fill the section of the pipe. 4 §

§ l.;f e;

L

FIG. 88. FIG. 89.

Let w be the section and 11 the velocity of the stream at bb. At aa the section will be ctw, and the velocity (w/c, w)v=v/cl, where eu is the coefficient of contraction. Then the head lost is fy... = (ff/C. -102/2g= (I/c.- 02112/2g; and, if cc is taken 0-64,

b, ,, =0-316112/2g. (2)

The value of the coefficient of contraction for this case is, however, not well ascertained, and the result is somewhat modified by friction. For water entering a cylindrical, not bell-mouthed, pipe from a reservoir of indehnitely large size, experiment gives D-=0~505 11”/21- (3)

If there is a diaphragm at the mouth of the pipe as in fig. 89, let wi be the area of this orifice. Then the area of the contracted stream is can, and the head lost is

be =l(w/¢¢w1)" I l”f/2/28

=s“.1»'/2g (4)

if § ', is put for {(w/c¢w1)-IP. Weisbach has found experimentally the following values of the coefficient, when the stream approaching the orifice was considerably larger than the orifice:- 6¢= .616 .614 .612 .610 .617 .605 .603 .GOI .SQ8 .596 = 231.7 50.99 19.78 9.612 5.256 3.077 1.876 1169 0.734 0.480 Vi.. ' H 4 .4 .., 4 4 4 4 4

When a diaphragm was placed in a tube of uniform section (fig. 90) FIG. 90.

the following values were obtained, wi being the area of the orifice and w that of the pipe:-

C0160-

/ - 0.1 0.2 0.3 0.4 0,5 0.6 0.7 0.8 0.9 1 0 C, = .624 .632 .643 .659 .681 .712 .755 .813 .892 I.O0 £c= 225.9 47.77 30.83 7.801 1.753 1.796 .797 .290 .060 .000 Elbojws.-Weisbach considers the loss of head at elbows (fig.91) to be due to a contraction formed by the stream. From experiments 9= 45° 50° 55° 60° 65° 70° 90° with a pipe Ii in. diameter, he found the loss of head I), =g-81,2/zg; (5) § , = 18-7 32-6 58-8 118 256 751 as fe =0'9457 S1H2i4>+2'047 S1114 i4>¢ “ 29° 40° 60° 80° 90° 100° 110° 120° 130° 140° I fe: ' 0.046 I Q139, 0.364 i o.740| o 984| I.26O| 1.5561 1.861 l 2.1581 2.431<| Hence at a right-angled elbow the whole head due to the velocity very nearly is lost. Bands.-Weisbach traces the loss of head at curved bends to a similar cause to that at <~ elbows, f but;tihe coeffi§ cients or ben s are not w very satisfactorily ascer tained. /Veisbach obw tained for the loss of hpad atl a bend in a pipe

i o circu ar section

b*-, ~;='°° f>»=fw2/zg; <6> § fb=o-131+1-847(d/2p)S, { where d is the diameter

of the pipe and p the

FIG. 91. radius of curvature of the bend. The resistance at bends is small and at present very ill determined. Valves, Cocks and Sluices.-These produce a contraction of the - water-stream similar to that for an abrupt 5 loss of head may be taken as before to be =§ ~-2 2g- where 1/ is the xelocity in the pipe beyond the valve and § ', a coefficient determined by experiment. The diminution of section already discussed. The F, fn, /, (7) v l following are Weisbach's results. Sluice in Pipe of Rectangular Section (fi . 2). FIG' 92' Section at s1uice=w1 in pipe=w. g 9 <-11/w= I-0 0-9 0-8 0-7 ~o-6 o-5 0-4 o-3 o-2 0-1 I 4 § '»= 0-oo -o9 -39 -95 2-08 4-02 8-12 17-8 44-5 193 Sluice in Cylindrical Pipe (fig. 93). Ratio of height of opening to diameter of pipe wi/w = 1 0 -3 i % iv § 1.00 0.948 .856 .740 .609 .466 .315 .159 ocoi 0.07 i o26' 0.81 | Loéi 5.52 I 17,0 '978 l § '., = / 3/ ww/4 /' 9 - - »:j; § t nw////////W, ff, ' g~§§ , ,, W”, W, /, W, % // / / / FIG. 93. FIG. 94. Cock in a Cylindrical Pipe (fig. 94). Angle through which cock is turned=0. 0= 5° 10° 15° 20° 25° 30° 35° Ratio of) c r o ss 2 -926 -850 ' -772 -692 -613 -535 -458 sections) § ¢= '05 '29 7| '75 V56 | 3'10 5'47 9°68 6? 409 V450 ' go l 55° 600 650 820 Ratio of c rto s s -385 -315 -250 -190 -137 'OQI o sec ions 5-, = 17-3 3I°2 52-6 IOC) 1-o6 486 so cal Pipe (fig. 95) Throttle Valve in a Cylindri If 0 = 5° 10° 15° I-1 = '24 '52 '90

0 20 25° 50° 35° 40° 1-54 i 2-51 3-91 i 6'22 iw-8 1 § 84. Practical Calculations the following explanations it will be assumed that the pipe is of so great a length that only the loss of head in friction against " on the Flow of Water in Pipes.-In the surface of the pipe needs WW/W H »-'JH / to be considered. In general M /M" /W ”W/W” it is one of the four quantities ' 3 d, i, 'U or Q which requires ' " '”"' ""“” to be determined. For since 'ii the loss of head h is given by /fr -W///2 <f- » '/M//7/7/W” //W the relation h=il, this need not be separately considered. FIG 95 There are then three equations (see eq. 4, § 72, and 9a, § 76) for the solution of such problems as arise:- i'=a(I'l'I/I2d)i (I) where a =o-005 for new and =o-01 for incrusted pipes. W2/22 = idl- (2) Q = i1fd2v~ (3) Problem 1. Given the diameter of the pipe and its virtual slope, to find the discharge Hlld velocity of flow. Here d and i are given, and Q and 'v are required. Find § ' from (I); then v from (2); lastly Q from (3). This case presents no difficulty. By combining equations (1) and (2), v is obtained directly:- U=/ (gdi/25) =/ (g/2¢)/ [di/ll +I/1211) 1- (4) For new pipes / (g/20.) =56-72 For incrusted pipes =4o-13 For pipes not less than 1, or more than 4 ft. in diameter, the mean values of gf' are For new pipes .. .. 0-00526 For incrusted pipes ..... O°01052. Using these values we get the very simple expressions=55-31/ (di) for new pipes 1 (46) =39-11/ (di) for incrusted pipes) Within the limits stated, these are accurate enough for practical purposes, especially as the precise value of the coefficient § ' cannot be known for each special case. Problem 2. Given the diameter of a pipe and the velocity of flow, to find the virtual slope and discharge. The discharge is given by (3); the proper value of of by (1); and the virtual slope by (2). This also presents no special difficulty. Problem 3. Given the diameter of the pipe and the discharge, to find the virtual slope and velocity. Find 11 from (3); § ' from (1); lastly i from (2). If we combine (1) and (2) we get i=§ '(v”/2§) (4/ll) =2¢1{I +I/I2fl}'v'/ed; (5) and, taking the mean values of § ' for pipes from 1 to 4 ft. diameter, given above, the approximate formulae are i=o-ooo3268 112/d for new pipes Q (5a) =o-0006536 vi'/d for incrusted pipes 5 Problem 4. Given the virtual slope and the velocity, to find the diameter of the pipe and the discharge. The diameter is obtained from equations (2) and (1), which give the quadratic expression d”-d(2av2/gi) -1.1212/6gi=0. d=11'v”/gi+/ l(fw”/gi) (<»'v”/gi+I/5)}~ (6) For practical purposes, the approximate equations d=2o.v2/gi+1/12 . (6a) =O'O0O3I oz/i+-083 for new pipes =O'OOO62 of/i-1—083 for incrusted pipes are sufficiently accurate. Problem 5. Given the virtual slope and the discharge, to find the diameter of the pipe and velocity of flow. This case, which often occurs in designing, is the one which is least easy of direct solution. From equations (2) and (3) we getd'=32s“Q”/g1f”i- (7) If now the value of f in (1) is introduced, the equation becomes very cumbrous. Various approximate methods of meeting the difficulty may be used. (a) Taking the mean values of g' given above for pipes of 1 to 4 ft. diameter we get 1l=%/(325/§ 1f“)i/(Qi/i) (8) =O'2216§ / (Q2/i) for new pipes =O'254I Q/ (Q2/i) for incrusted pipes; equations which are interesting as showing that when the value of 5° is doubled the diameter of pipe for a given discharge is only increased by 13 %. (b) A second method is to obtain a rough value ofdby assuming § '=a.. This value is

d' =i/ (32Q'/g1f'i)%/ a =0-6319 if (Q”/i>'/ a-Then a ver) approximate value of 3' is

§ '=<=<1+1/12d'>;

and a revised value of d, not sensibly differing from the exact value, IS

fi” =i/ (32Q2.i§ "'2i)§ / I' =0°6319 V (Q'/DV I'(c) Equation 7 may be put in the

if the average demand is 25 gallons per head per day, the mains should be calculated for 50 gallons per head per day. § 86. Determination of the Diameters of Dijferent Parts of a Water Main.-When the plan of the arrangement of mains is determined upon, and the supply to each locality and the pressure required is ascertained, it remains to determine the diameters of the pipes. Let Hg. 97 show an elevation of a main ABCD. . ., R being the reservoir from which the supply is derived. Let NN be the datum line of the levelling operations, and Hn, Hz, ...the heights of the main above the datum line, Hi- being the height of the water surface in the form Q

d=§ /(32aQ2/g1r2i)%' (I +1/12d). (9) , § ; -, ? -'~'~'H~2::; ° ';::fr' '°" “ ' """ """ ""' “"' °""""""°" Expanding the term in brackets, 1”'*'- """"~3§ -=, ?;f: @.... ' & 6 W (1 +1/1 zd) = 1 -i-I/60d-I/ISOO112... -B 11 d, i ""' Slowking4Farmbrough's revenge. ' T" A Neglecting the terms after the second, ' I 'Q ' Qz@~';' 'B: d = 2/ (321, /gw2)v (Q2/i).{ 1 + 1 /sed; g XYZ* nz” '-~s = V (32¢1/gr”)§ ~/(Q”/1)+0-01667;(9r1): = ' 51, '~-and L, - of l 5 '~~ 0

M(32U.:/g .2) =o-219 for new pipes ' 5 Z i3”" f " 311% N T =O'252fOI'iI'1CI'l1St€dpip€S. Jd3Q.?T{;-E ':'T'EH"* § 8'. Arrangement of Water Mains 5 ' ' -64: for To:'iv11.s' Suppl .-Town mains are - — .v.....!Z.... .H ""P...~.I!!?!'.l'l'-»~.»—¥»-»~~—~»- —"—'~~°'”"“"! °"" "°' ' ""'°“ °°"' '°~-5-“*°° usually suppliediiy gravitation from

a service reservoir, which in turn is

supplied by gravitation from a storage reservoir or by pumping from a lower level. The service reservoir should contain three days' supply or in important cases much more. Its elevation should be such that water is delivered at a pressure of at least about loo ft. to the highest parts of the district. The greatest ressure in the mains is usually about 200 ft., the pressure for which ordinary pines and fittings are designed. Hence if the district supplied has l*” “"1:7H' L J Z .. .. . .-4-

-1- 49" eve one I A(

s if "I 0

I

'=91 |<¢!. Q' 'l9;T'l .' , ' "“i

1' 'V I

A | .?: -5 4

“L mf 2?° "

s ' 1

aw “', "

f I £56

I m y ILO, !

~ 4 1

56:4

L . . . . . i . . L . . . ... i . .. FIG. 96.

great variations of level it must be divided into zones of higher and lower pressure. Fig. 96 shows a district of two zones each with its service reservoir and a range of pressure in the lower district from 100 to 200 ft. The total supply required is in England about 25 gallons per head per day. But in many towns, and especially in America, the supply is considerably greater, but also in many cases FIG. 98. B V K V

reservoir from the same datum. Set up next heights AA), BB1, '... representing the minimum pressure height necessary for the adequate supply of each locality. Then A1B1C1D1.. . is a line which should form a lower limit to the line of virtual slope. Then if heights ba, bb, bc... are taken representing the actual losses of head in each length la, lb, l¢... of the main, AUBOCO will be the line of virtual slope, and it will be obvious at what points such as D0 and EU, the pressure is deficient, and a different choice of diameter of main is required. For any point z in the length of the main, we have 2 Pressure height=H, -H, -(fy, -|~f)1, +...§ ,). Where no other circumstance limits the loss of head to be assigned to a given length of main, a consideration of the safety of the main from fracture by hydraulic shock leads to a limitation of the velocity of flow. Generally the velocity in water mains lies between 1% and 4% ft. per second. Occasionally the velocity in pipes reaches 10 ft. per second, and in hydraulic machinery working under enormous pressures even 20 ft. per second. Usually the velocity diminishes along the main as the discharge diminishes, so as to reduce somewhat the total loss of head which is liable to render the pressure insufficient at the end of the main.

]. T. Fanning gives the following velocities as suitable in pipes for towns' supply:»-Diameter

in inches 4 8 12 18 24 30 36

Velocity in feet per sec. 2-5 3-0 3-5 4-5 5-3 6-2 7-o § 87. Branched Pipe connecting Reservoirs at Different Levels.-Let A, B, C (fig. 98) be three reservoirs connected by the arrangement of pipes shown, -ll, dl, QI, 111; lg, 112, Q2, 112; 13, da, Q3, '03 being the length, diameter, discharge and velocity in the three portions of the main pipe. Suppose the dimensions and positions of the pipes known and the discharges required.

If a pressure column is introduced at X, the water will rise to a height XR, measuring the pressure at X, and aR, Rb, Rc will be the lines of virtual slope. If the free surface level at R is above b, the reservoir A supplies B and C, and if

. § ; T g, rg —— ..- ~- . <-, —~-~- -T-~-~-»———~.q——»=~»- —=¢———= — r»— R is below b, A and B supply C.

% Q Consequently there are three cases 1-7§ ";T "l ~§ &"f § § § } I. R above b; Q1=Q2-I-Q3. R i ¥ Q 5 II. R level with b; Q1=Q3; Q2=o f A f Zm Q 5- 5 111. Rbe1Qwb;Q, +Q, =Q, L 20 i' ' 'Ui 1 : 5; < Ego To determine which case has to be t i 'Q dealt with in the given conditions, 5 ' " A “' 5 5 E Q suppose the pipe from X to B closed Hr: § ; ~-, ;C, : , Q; by a sluice. Then there is a simple

2 Q . » is I ~, F main, and the height of free surface

1 ' 3 I E aB; ' -' 5;0. '%E~-;% h' at X can be determined. For this ffm; = ;, Q is condition

2; Ha; T i In-h'=r<vf/2g><4li/do,

4: 5 » 5 II 5 D; E " =32§ “Q'l1/g1r'di5; Q 2 fe 2; -F 1-'-fi=f<»-2/2g><41-/dg., ,,

- J. - .131 . § ' . . L .. = . 5 ' Q i =32s' Sgr- 3i

—Y —— —- -~~»- 71 —ff- ——~ 35 — ~J-:!— —~—~.~—5—»~i»-~...-~...-5 .... - where Q' is the common discharge N N of the two portions of the pipe.

FIG. 97. Hence

a good deal of the supply is lost by leakage of the mains. The su ply through the branch mains of a distributing system is calculated liiom the population supplied. But in determining the capacity of the mains the Euctuation of the demand must be allowed for. It is usual to take the mximum demand at twice the average demand. Hence (he -W)/(h'-he) =lid3“/lid1',

from which h' is easily obtained. If then h' is greater than hb. opening the sluice between X and B will allow flow towards B, and the case in hand is case I. If h' is less than hb, opening the sluice will allow flow from B, and the case is case III. If h'=h, ,, the case is case II., and is already completely solved. The true value of h must lie between h' and hr. Choose a new value of h, and recalculate Qi, Q2, Q3. Then if Q1>Q¢~|-Q3 in case I.,

or Q1+Q2>Q3IH C356 III.,

the value chosen for li is too small, and a new value must be chosen. If

Q1<Q¢-l-Q3 in case I.,

or Q1-}-Qg<Q3 in case III.,

the value of h is too great.

Since the limits between which li can vary are in practical cases not very distant, it is easy to approximate to values sufficiently accurate. § 88. Water Hammer.-If in a pipe through which water is Howing a sluice is suddenly closed so as to arrest the forward movement of the water, there is a rise of pressure which in some cases is serious enough to burst the pipe. This action is termed water hammer or water ram. The fluctuation of pressure is an oscillating one and gradually dies out. Care is usually taken that sluices should only be closed gradually and then the effect is inappreciable. Very careful experiments on water hammer were made by N. ]. joukowsky at Moscow in 1898 (Stoss in Wasserleilungen, St Petersburg, 1900), and the results are generally confirmed by experiments made by E. B. Weston and R. C. Carpenter in America., Ioukowsky used pipes, 2, 4 and 6 in. diameter, from 1000 to 2500 ft. in length. The sluice closed in 0~03 second, and the fluctuations of pressure were automatically registered. The maximum excess pressure due to water hammer action was as follows:-

V Pipe 4-in. diameter. V Ihpe 6'ifT. dianieter. Velocity Excess Pressure. Velc;;ity Excess Pressure. ft. per sec. Tb per sq. in. ft. per sec. Tb per sq. in. 0-5 31 i 0-6 43

2-9 168 3-0 173

4-1 232 I 5~6 369

t 9-5- l .- Se. - l . 1 L is

In some cases, in fixing the thickness of water mains, 100 lb per sq. in. excess pressure is allowed to cover the effect of water hammer. With the velocities usual in water mains, especially as no valves can be quite suddenly closed, this appears to be a reasonable allowance (see also Carpenter, Am. Soc. Mech. Eng., 1893).

IX. FLOW OF COMPRESSIBLE FLUIDS IN PIPES

§ 89. Flow of A ir 'in Long Pipes.-'Nhen air flows through a long pipe, by far the greater part of the work expended is used in overcoming frictional resistances due to the surface of the pipe. The work expended in friction generates heat, which for the most part must be developed in and given back to the air. Some heat may be transmitted through the sides of the pipe to surrounding materials, but in experiments hitherto made the amount so conducted away appears to be very small, and if no heat is transmitted the air in the tube must remain sensibly at the same temperature during expansion. In other words, the expansion may be regarded as isothermal expansion, the heat generated by friction exactly neutralizing the cooling due to the work done. Experiments on the pneumatic tubes used for the transmission of messages. by R. S. Culley and R. Sabine (Proc. Inst. Civ. Eng. xliii.), show that the change of temperature of the air fiowing along the tube is much less than it would be in adiabatic expansion.

§ 90. Diferential Equation of llze Sleady Motion of Air Flowing in a Long Pipe of Uniform Section.-When air expands at a constant absolute temperature 1, the relation between the pressure p in pounds per square foot and the density or weight per cubic foot G is given by the equation

p/G = cr, (1)

where c =53~ 15. Taking r =52I, corresponding to a temperature of 60° Fahr.,

cr =27690 foot-pounds. (2)

The equation of continuity, which expresses the condition that in steady motion the same weight of fluid, W, must pass through each cross section of the stream in

" V . the unit of time, is

ii Q Q G§ lu=W=constant, (3)

{' f ¢; , ,; E where Q is the section of the E 5;: pipe and u the velocity of

Q Q } 1 the air. Combining (1) and

'::: (3).

7, Qup/;=c1==constant. (3a)

4° A” A1 A1 Since the work done by

FI gravity on the a1r during its

G 99 flow through a pipe due to

variations of its level is generally small compared with the work done by changes of pressure, the former may in many cases be neglected.

Consider a short length dl of the pipe limited by sections Au, A1 at a distance dl (ng. 99). Let p, u be the pressure and velocity at Ae, p-f-dp and u+du those at Al. Further, suppose that in a very short

time dt the mass of air between AOA, comes to A'0A', so that AUAQ; = nd! and A1A'1 = (ii-I-dn)dt1. Let SZ be the section. and ni the hydraulic mean radius of the pipe, and W the weight of air flowing through the pipe per second.-From

the steadiness of the motion the weight of air between the sections A0A'0, and A1A'1 is the same. That is, Wd! = Giludt = G§ 2(u+du)dt.

By analogy with liquids the head lost in friction is, for the length dl (see § 72, eq. 3), § '(u2/2g)(dl/m). Let H=u2/zg. Then the head lost is § '(H/m)dl; and, since 'Writ Tb of air How through the pipe in the time considered, the work expended in friction is -§ '(H/m)Wdl dt. The change of kinetic energy in di seconds is the difference of the kinetic energy of A0A'0 and AIAQ, that is, (W/g)dt{ (u -I-dn)2-~u2}/2 == (W/g) ndu dt =VdHdt. The work of 'expansion when Sludt cub. ft. of air at a pressure p expand to § 2(u+du)d! cub. ft. is (lp dudt. But from (3a) u=crW/Up, and therefore

du/dp = -c1-N/S2p2.

And the work done by expansion is- (cv-W/p)dp di. The work done by gravity on the mass between A0 and A1 is zero if the pipe is horizontal, and may in other cases be neglected without great error. The Work of the pressures at the sections AUA, is pfludt- (Pd-dp)§ Z(u+du)di

= - (pdu+ndp)Qdt

But from (ga)

pu = constant,

pdn+ndp = o,

and the work of the pressures is zero. Adding together the quantities of work, and equating them to the change of kinetic energy, WdI-ldt = - (UVV/p)dp di - § '(H/m)'V dl dt dH+ (61/P)dP+$“(H/m)dl =0,

dH/H -1- (cr/Hp)dp+§ 'dZ/in = 0 (4)

But u =c1-W/Stp,

and H =u2/2g=c”r2W/2g£2*p2,

dH/H+(2gSl'p/c'rV'*)dp+§ dI/m =0. (411) For tubes of uniform section m is constant; for steady motion W is constant; and for isothermal expansion -r is constant. Integrating, log H +gQ2p2/VX/“cr -i"§ 'l/111. = constant; (5) for l=o, let I'I=Hn, and;b=p0;

and for l=i, let l-I=H1, and p=-pi.

105; (Hi/Ir1oH-(392/'Wvf) (PF -Pug) +C 1/"1 = 0~ (511) where po is the greater pressure and pl the less, and the flow is from Ao towards Al.

By replacing W and H,

l0s(1><>/Pi)+(gvf/'uo”l>o”>(PF -Poi) + 5 1/"1 =0~ (6) Hence the initial velocity in the pipe is Ho = w/ ligCT(P0“-i>12)i/iP02(§ l/"1+l0s(Po/1bi)} l~ (7) V'hen I is great, log po/pl is comparatively small, and then l¢o=/[(§ C'fm/fl)i(Po2"P12)/Z'02ll» (761) a very simple and easily used expression. For pipes of circular section m =d/4, where d is the diameter?- uv = V [(1;CTfif"4§ 'l)i(P®”-PF)/P02il S (771) or approximately

no =(1-1319 -<>~7264;f>i/1>f>)~' (gvfd/4t1> (76) § 91. Coefficient of Friction for A'ir.~A discussion by Professor Unwin of the experiments by Culley and Sabine on the rate of transmission of light carriers through pneumatic tubes, in which there is steady flow of air not sensibly affected by any resistances other than surface friction, furnished the value § '= -007. The pipes were lead pipes, slightly moist, 2§ in. (0~187 ft.) in diameter, and in lengths of 2000 to nearly 6000 ft.

In some experiments on the flow of air through cast-iron pipes A. Arson found the coefficient of friction to vary with the velocity and diameter of the pipe. Putting

I = 11/1'-l-ff.

he obtained the following values-Diameter of Pipe I 5” for IOO ft.

in feet. °' B per second.

1~64 -00129 -00483 -00484

I 1 -07 -009 72 00640 -006 50

-83 -01525 00704 ~00719

-538 -03604 '0OQ4I -00977

-266 -03790 -00959 ~0<>997

-164 -04518 -or 167 ~OI2I2

It is worth while to try if these numbers can be expressed in the form proposed by Darcy for water. For a velocity of 100 ft. per second, and without much error for higher velocities, these numbers agree fairly with the formula

§ '=0'005(I+3/10fl)- is!

which only differs from Darcy's value for water in that the second term, which is always small except for very small pipes, is larger. Some later experiments on a very large scale, by E. Stoclcalper at the St Gotthard Tunnel, agree better with the value f=O'0028(I +3/rod). These pipes were probably less rough than Arson's. W'hen the variation of ressure is very small, it is no longer safe to neglect the variation of) level of the pipe. For that case we may neglect the work done by expansion, and then Zo-Zi'°Po/Go-P1/G1'ffvi/28) (l/m)=0, (10) precisely equivalent to the equation for the flow of water, zo and zi being the elevations of the two ends of the pipe above any datum, po and pi the pressures, G0 and Gi the densities, and 'U the mean velocity in the pipe. This equation may be used for the flow of coal gas. § 92. Distribution of Pressure in a Pipe in which Air is Flowing.-From equation (7a) it results that the pressure p, at l ft. from that end of the pipe where the pressure is po, is p=p0/{1 -§ 'lu0'/mgc-r }; (II) which is of the form 1>=~/(GH-12) for any given pipe with given end pressures. The curve of free surface level for the pipe is, therefore, a parabola with horizontal axis. Fig. loo shows calculated curves of pressure for two of Sabine's experiments, in one of which the pressure was greater than atmo-54- so 7 P . 3 +449 1, Hurt -io-0: 1:m;z9~1.s) A I I Q3-507 l' L J/

zo-oo

27-1.5. [1 za -as U C” 'Un (M-2.5)::,0 27 1-5 7-7

- .; - . 4 . | . - . N V J/ V o zna-sift. ~1-227Ft- sslo-.sF¢. sas-1-Ft. FIG. 100. spheric pressure, and in the other less than atmospheric pressure. The observed pressures are given in brackets and the calculated pressures without brackets. The pipe was the pneumatic tube between Fenchurch Street and the Central Station, 2818 yds. in length. The pressures are given in inches of mercury. Variation of Velocity in the Pipe.~Let po, ua be the pressure and velocity at a given section of the pipe; 17, u, the pressure and velocity at any other section. From equation (3a.) up = ci-W/Q =constant; so that, for any given uniform pipe, up=u0p0, u=4».f>», {1>; <12> which gives the velocity at any section in terms of the pressure, which has already been determined. Fig. IOI gives the velocity 4:-¢ C I 0 Vg/:w»""' |33:°”'7 284 .1 R, ,-nn |21

oe, zo-B 1

I nos Q 22 2 i, I I 1 1 |:

.

I I Q I ' I 'I »

I 5 3 g I

A 5 2113-511, 42 27 Fa sa-me-.sFe. 04~;4~f}-FIG. 101. curves for the two experiments of Culley and Sabine, for which the pressure curves have already been drawn. It will be seen that the velocity increases considerably towards that end of the pipe where the pressure is least. § 93. 'Weight of Air Flowing per Second.—The weight of air discharged per second is (equation 3a)-W = szuupo/Cf. From equation (7b), for a pipe of circular section and diameter d, W =i1r~/ lfzdtfeoz-Plz)/tlvfl. A I = -61 Ix' {li°(Po2-P12)/S'lTl- (13) foxlfrlat pp e Y W= <-691%—44@f=i».> <d=~/clot. use § 94. Application to the Case of Pneumatic Tubes for the Transmission of Messages-In Paris, Berlin, London, and other t 0wns, it has been found cheaper to transmit messages in pneumatic tubes than to telegraph by electricity. The tubes are laid underground with easy curves; the messages are made into a roll and placed in a light felt carrier, the resistance of which in the tubes in London is only % oz. A current of air forced into the tube or drawn through it propels the carrier. In most systems the current of air is steady and continuous, and the carriers are introduced or removed without materially altering the How of air. Time of Transit through the T ube.-Puttingt for the time of transit from 0 to l, Z t= odl/u, From (4a) neglecting dH/H, and putting m=d/4, dl = goliltpdp/2§ 'W2cr. From (I) and (3) u = Wc'r/pil; dl/ u = gdS't3p2dp/2§ W3c'1-2; z= "{f§ ';gdn3p2dp/2;WSr2f2, = £5123(1903-P13)/6§ WC'f? (14) But /V = Pougfl/CT; °. If =gd€T(P03-P13)/6§ 'P031t0?, U = s'5l¥(P03*P1“) /6 (35711) f(P02-Plz) 2- (15) If 1 =521°, corresponding to 60° F., f='001412S'5l3(P03'P13)/di=(P02*P12li5§ (150) which gives the time of transmission in terms of the initial and final pressures and the dimensions of the tube. Mean Velocity of Transmission.-The mean velocity is l/t; or, for -r = 52 r °, “mean =0'708(ld(P02°'P12)g/flipoa p13)i The following table gives some results:- I, , T Mean Velocities for Tubes of a. Ib per Sq in length in feet. Y po pi 1000 2000 3000 4000 5000 Vacuum - - 15 5 99'4 70'3 57-4 49-7 44-5 Working . l 15 IO 07-2 47-5 38-8 34-4 30-x P . 20 15 57-2 40~5 33-0 28-6 25-6 25 15 746 52'7 434 37-3 35's - ~ 30 15 347 60'0 4970 42'4 37'9 Limiting Velocity in the Pipe when the diminished indefinitely.-If in the last equation there be put pi =0, then Pressure at one End is 1»£'mean=0'708/ where the velocity is independent of the pressure po at the other end, a result which apparently must be absurd. Probably for long pipes, as for orifices, there is a limit to the ratio of the initial and terminal pressures for which the formula is applicable. X. FLOW IN RIVERS AND CANALS § 95. Flow of Water in Open Canals and Rivers.-When water Hows in a pipe the section at any point is determined by the form of the boundary. When it Hows in an open channel with free upper surface, the section depends on the velocity due to the dynamical conditions. Suppose water admitted to an unfilled canal. The channel will gradually fill, the section and velocity at each point gradually changing. But if the inflow to the canal at its head is constant, the increase of cross section and diminution of velocity at each point attain after a time a limit. Thenceforward the section and velocity at each point are constant, and the motion is steady, or permanent regime is established. If when the motion is steady the sections of the stream are all equal, the motion is uniform. By hypothesis, the inH0w (lv is constant for all sections, and Q is constant; therefore must be constant also from section to section. The case is then one of uniform steady motion. In most artificial channels the form of section is constant, and the bed has a uniform slope. In that case the motion is uniform, the depth is constant, and the stream surface is parallel to the bed. If when steady motion is established the sections are unequal, the motion is steady motion with varying velocity from section to section. Ordinary rivers are in this condition, especially where the flow is modified by weirs or obstructions. Short unobstructed lengths of a river may be treated as of uniform section without great error, the mean section in the length being put for the actual sections. In all actual streams the different fluid filaments have different velocities, those near the surface and centre moving faster than those near the bottom and sides. The ordinary formulae for the flow of streams rest on a hypothesis that this variation of velocity may be neglected, and that all the filaments may be treated as having a common velocity equal to the mean velocity of the stream. On this hypothesis, a plane layer abab (fig. 102) between sections normal to the direction of motion is treated as sliding down the channel to a'a'b'b' without deformation. The component of the weight parallel to the channel bed balances the friction against the channel, and in estimating the friction the velocity of rubbing is taken to be the mean velocity of the stream. In actual streams, however, the velocity of rubbing on which the friction depends is not the mean AULICS 69

variation of the coefficient of friction with the velocity, proposed an expression of the form

§ =<1(l +3/U), (5)

and from 255 experiments obtained for the constants the values 0. =0-007409; li =0- 1920.

This gives the following values at different velocities 2v- 0-3 0-5 0-7 1 15 2 3 5 7 10 I5

g'= 0~01215 0-01025 0-00944 0-00883 0-00836 O°OO8I2 0-90788 0-00769 0'OO76I 0-00755 0-00750 l

velocity of the stream, and is not in any simple relation with it, for ' In using this value of § ' when 11 is not known, it is best to proceed channels of different forms. The by approximation. Q 6 theory is therefore obviously based § 98. Darcy and Bazirfs Expression for the Coejicient of Friction.-;, , I on an imperfect hypothesis. How- Darcy and BRZIHYS researches have sh0w11 that I varies very greatly Q 5 ever, by taking variable values for for different degrees of roughness of the channel bed, and that it WW qzj- /5 5 ' the coefficient of friction, the errors also varies with the dimensions of the channel. They give for I an %@ 0/, E of the ordinary formulae are to a empirical expression (similar to that for pipes) of the form %/y M' great extent neutralized, and they 5-:au +5/my (6) are/4...

may be used without leading to

practical errors. Formulae have

been obtained based on less represent they are not practically so

reliable, and are more complicated than the formulae obtained in the manner described above.

§ 96. Steady Flow of Water with Uniform Velocity in Channels of Constant Section.-Let ao/, bb' (fig. 103) be two cross sections normal to the direction of motion at a distance dl. Since the mass aa'bb moves uniformly, the external forces acting on it are in equilibrium Let S2 be th f

e area 0 the cross sections, X the wetted perimeter, Fic.. 102.

stricted hypotheses but at

g ... .... ........ .. . . .-.v

f =1

where rn is the hydraulic mean depth. For different kinds of channels they give the following values of the coefficient of friction 1-7 I Kind of Channel. | 0., s

cement or planed timber ...

Smooth channels, sides of ashlar, brickwork, planks .......

Rough channels, sides of rubble masonry or pitched with stone

IV. Very rough canals in earth ..... V. Torrential streams encumbered with detritus 0-00294 0-10

II.

III.

0-00373 0-23

I. Very smooth channels, sides of smooth


FIG. 103.

pq-l-qr-l-rs, of a section. Then the quantity m=S2/X is termed the hydraulic mean depth of the section. Let 'v be the mean velocity of the stream, which is taken as the common velocity of all the particles, i, the slope or fall of the stream in feet, per foot, being the ratio bc/ab.

The external forces acting on aa/bb' parallel to the direction of motion are three:-(a) The pressures on aa' and bb', which are equal and opposite since the sections are equal and similar, and the mean pressures on each are the same. (b) The component of the weight W of the mass in the direction of motion, acting at its centre of gravity g. The weight of the mass aa'bb' is GS2:ll, and the component of the weight in the direction of motion is GSZdl>< the cosine of the angle between Wg and ab, that is, Gfldl cos abc=G§ ?dl bc/ab-Gftidl. (c) There is the friction of the stream on the sides and bottom of the channel. This is proportional to the area Xdl of rubbing surface and to a function of the velocity which may be written f(1'): f(v) being the friction per sq. ft. at a velocity 11. Hence the friction is -xdl f('v). Equating the sum of the forces to zero, GQi dl-Xdl f(v) =0,

f(°u)/G=§ Zi/X=mi. (1)

But it has been already shown (§ 66) thatf(v) = § 'G1J2/2g, (112/2g =mi. (2)

This may be put in the form

v = / (28/DV (mi) =v~/ ( H); (211)

where c is a coefficient depending on the roughness and form of the channel.

The coefficient of friction f varies greatly with the degree of roughness of the channel sides, and somewhat also with the velocity. It must also be made to depend on the absolute dimensions of the section, to elirnin. t th f l ' 1 ' ' 1 " 1 e e error 0 neg ecting the xarlations of xelocity in the cross section. A c0mm0n mean value assumed for I is 0-00757. The range of values will be discussed presently. It is often convenient to estimate the fall of the stream in feet per mile, instead of in feet per foot. Iff is the fall in feet per mile, f= 5280 i.

Putting this and the above value of of in (20), we get the very simple and long-k110wn approximate formula for the mean velocity of a stream-

”=1l%/(2H1f)- (3)

The How down the stream per second, or discharge of the stream, is Q=§ Z'v=9cx/ (mi). (4)

§ 97. Coejicient of Friction for Oben Cliannels.-Various expressions have beeu proposed for the coemcient of friction ior| channels as for pipes. Weisbach, giving attention chiefly to the The last yalues (Class V.) are not Darcy and Bazin's, but are taken from experiments by Ganguillet and Kutter on Swiss streams. The following table very much facilitates the calculation of the mean velocity and discharge of channels, when Darcy and Bazin's value of the coefficient of friction is used. Taking the general formula for the mean velocity already given in equation (2a) above, 1; =c/ (mi),

where c=/ (2g/5), the following table gives values of c for channels of different degrees of roughness, and for such values of the hydraulic mean depths as are likely to occur in practical calculations:- Valuex ofc in v = cy/ (mi), cleflucedfrovn Darcy and Bazin's Values.

Q.; gd; om 52 -57.5 Tggn =§ - E um 'Sf .555 'gg e * E “ -se -2' -.1 =-'= e ., J*, -5 “ ~ 2 m gd 2°-3 E2 1153 'go;=>§ 3; E15 § '§ $5 = "5 £5 -iii 80° 6 '5 as £5 eil 55 § “' < §

-25 125 95 57 26 18-5 8-5 147 130 112 89 . -55 135 112 Q2 36 25-2 9-3 147 130 112 90 71 -7 139 Il 1 42 30- 9- 147 I3O II2 90 . 1-1; 141 119 87 34-9 10-0 147 130 112 QI 72 1- 143 122 94 41-2 II 147 I3O 113 92 . 2-0 144 124 98 62 46°C 12 147 130 II3 93 74 2-5 145 122 101 37 13 147 130 II3 94 . 3-0 145 12 104 0 3 14 147 130 II3 95 .. 3-5 146 127 105 73 . 15 147 130 114 96 77 4-0 146 128 IO6 76 58 16 147 130 II4 97 .. 4-5 146 128 107 78 . 17 147 130 114 97 .. 5-0 146 128 IO8 80 62 18 147 130 II4 98 .. 5-5 146 129 109 82 20 147 131 II4 98 80 6-0 147 IZQ 110 84 65 25 148 ISI 115 100 . 6-5 147 129 110 85 . 30 148 I3I IIS 102 83

7-0 147 I2Q 110 86 67 40 148 I3I 116 103 85

7-5 147 129 III 87 50 148 131 II6 104 86 L8-0 147 I3O III 88 69 oo 5348 131 ' 117 ' 108 591 § 99. Gangnillet and Kutter's Modi ed Darcy' Formula.-Startin 3

§ from the general expression z1=c/ rni, Ganguillet and Kutter examined the variations of C for a wider variety of cases than those discussed by Darcy and Bazin. Darcy and Bazin's experiments were confined to channels of moderate section, and to a limited variation of slope. Ganguillet and Kutter brought into the discussion two very distinct and important additional series of results. The gaugings of the Mississippi by A. A. Humphreys and H. L. Abbot afford data of discharge for the case of a stream of exceptionally large section and OI very row slope. Un the otner hand their li own measurements of the flow in the regulated channels of some detritus

Swiss torrents gave data for cases in which the inclination and roughness of the channels were exceptionally great. Darcy and Bazin's experiments alone were conclusive as to the dependence of the coefficient c on the dimensions of the channel and on its roughness of surface. Plotting values of c for channels of different inclination appeared to indicate that it also depended on the slope of the stream. Taking the Mississippi data only, they found c=256 for an inclination of 0-0034 per thousand,

154- u rv O'02 vv

so that for very low inclinations no constant value of c independent of the slope would furnish good values of the discharge. In small rivers, on the other hand, the values of c vary little with the slope. As regards the influence of roughness of the sides of the channel a different law holds. For very small channels differences of roughness have a great influence on the discharge, but for very large channels different degrees of roughness have but little influence, and for indefinitely large channels the influence of different degrees of roughness must be assumed to vanish. The coefficients given by Darcy and Bazin are different for each of the classes of channels of different roughness, even when the dimensions of the channel are infinite. But, as it is much more'pr0bable that the influence of the nature of the sides diminishes indennitely as the channel is larger, this must be regarded as a defect in their formula. Comparing their own measurements in torrential streams in Switzerland with those of Darcy and Bazin, Ganguillet and Kutter found that the four classes of coefficients proposed by Darcy and Bazin were insufficient to cover all cases. Some of the Swiss streams gave results which showed that the roughness of the bed was markedly greater than in any of the channels tried by the French engineers. It was necessary therefore in adopting the plan of arranging the different channels in classes of approximately similar roughness to increase the number of classes. Especially an additional class was required for channels obstructed by detritus. To obtain a new expression for the coefficient in the formula v= / (23/s“)~' (mi) =¢/ (mi).

C-anguillet and Kutter proceeded in a purely empirical way. They found that an expression of the form

0 = <1/(I +6/V 111)

could be made to lit the experiments somewhat better than Darcy's expression. Inverting this, we get

I/C = 1/<1+B/a/ m.

an equation to a straight line having I//m for abscissa, I/c for ordinate, and inclined to the axis of abscissae at an angle the tangent of which is B/a.

Plotting the experimental values of 1/c and 1// m, the points so found indicated a curved rather than a straight line, so that B must depend on o.. After much comparison the following form was 0 = (AH/11)/(I +A#/V M).

where 71 is a coefficient depending only on the roughness of the sides of the channel, and A and I are new coefficients, the value of which remains to be determined. From what has been already stated, the coefficient c depends on the inclination of the stream, decreasing as the slope i increases.

Let A = a-l-p/i.

Then c=(a-l-l/n-I-p/1)/{1+(a+p/z)n/»/ m}, the form of the expression for c ultimately adopted by Ganguillet and Kutter.

For the constants a, l, p Ganguillet and Kutter obtain the values 23, 1 and 0-00155 for metrical measures, or 41-6, I'8II and 0'OO28I for English feet. The coefficient of roughness n is found to vary from 0-008 to 0-050 for either metrical or English measures. The most practically useful values of the coefficient of roughness n are given in the following table:-

arrived at-Nature

of Sides of Channel. Coefficient of

Roughness n.

Well-planed timber ..... 0-009

Cement plaster .. .... 0-010

Plaster of cement with one-third sand . 0-01 1 Unplaned planks ...... . 0-012

Ashlar and brickwork ..... 0-01 3

Canvas on frames 0-015

Rubble masonry .......... 0'0I 7

Canals in very firm gravel .... .. 0-020 Rivers and canals in perfect order, free from stones 002 or weeds .. ........ 5

Rivers and canals in moderately good order, not quite free from stones and weeds .... 0030 Rivers and canals in bad order, with weeds and detritus .... ..... . 0.035

Torrential streams encumbered with detritus 0-050 Ganguillet and Kutter's formula is so cumbrous that it is difficult to use without the aid of tables.

Lowis D'A. lackson published complete and extensive tables for facilitating the use of the Ganguillet and Kutter formula (Canal and Culvert Tables, London, 1878). To lessen calculation he puts the formula in this form:-

M =u(41-6+0-00281/l);

1/= (V M/"WM +I—311)/(M~l-V m)l/ (mil

The following table gives a selection of values of M, taken from Iacks0n's tables:-

C 3 Values of M for n-

o-010 0-012 0-015 0-017 0-020 0-025 0-030 3

-00004 1-1185 1-3422 1-6777 1-9014 3

-00006 0-8843 1-0612 1-3264 1-503 1-7686 2-210 2-6529 0 0

3555

3 7

-00008 0-7672 0-9206 1-1508 1-3042 1-5344 1-9180 2-3016 OOOIO 0-6970 0-8364 I-0455 1-1849 1-3940 1-7425 2-0910 -00025 0-5284 0-6341 0-7926 0-8983 1-0568 1-3210 1-5852 -00050 0-4722 0-5666 0-7083 0-8027 0-9444 1-1805 1-4166 -00075 0-4535 0-5442 -6802 -7709 0-9070 1-1337 1-3605 4 OO100 0-4441 0-5329 0-6661 0-7550 0-8882 1-1 102 1-3323 002O0 O'4300 0-5160 0-6450 0-7310 0-8600 1-0750 1-2900 -00300, O-4254 0-5105 0-6381 0-7232 0-8508 -0635 1-2762 -00001 3-2260 3-8712 4-8390 5-4842 6-4520 8-0650 K9-6780 -00002 1-8210 2-1852 2-7315 -0957 3-6420 4-5525 5463O 2

-2370 27962

1

A difficulty in the use of this formula is the selection of the coefficient of roughness. The difficulty is one which no theory will overcome, because no absolute measure of the roughness of stream beds is possible. For channels lined with timber or masonry the difficulty is not so great. The constants in that case are few and sufficiently defined. But in the case of ordinary canals and rivers the case is different, the coefficients having a much greater range. For artificial canals in rammed earth or gravel n varies from 0-0163 to O'O30I. For natural channels or rivers n varies from 0-020 to 0-035. In Jackson's opinion even Kutter's numerous classes of channels seem inadequately graduated, and he proposes for artificial canals the following classification:-

I. Canals in very firm gravel, in perfect order n =O'O2 II. Canals in earth, above the average in order n =O'O225 III. Canals in earth, in fair order ... n =O'O25 - IV. Canals in earth, below the average in order n = o-0275 V. Canals in earth, in rather bad order, partially overgrown with weeds and obstructed by n =0-03 Ganguillet and Kutter's formula has been considerably used partly from its adoption in calculating tables for irrigation work in India. But it is an empirical formula of an unsatisfactory form. Some engineers apparently have assumed that because it is complicated it must be more accurate than simpler formulae. Comparison with the results of gaugings shows that this is not the case. The term involving the slope was introduced to secure agreement with some early experiments on the Mississippi, and there is strong reason for doubting the accuracy of these results. § 100. Bazin's New Formula.-Bazin subsequently re-examined all the trustworthy gaugings of flow in channels and proposed a modification of the original Darcy formula which appears to be more satisfactory than any hitherto suggested (Etude ¢l'une noufvelle formule, Paris, 1898). He points out that Darcy's original formula, which is of the form mi/z12=a'l°/3/m, does not agree with experiments on channels as well as with experiments on pipes. It is an objection to it that if m increases indefinitely the limit towards which mi/1.12 tends is different for different values of the roughness. It would seem that if the dimensions of a canal are indefinitely increased the variation of resistance due to differing roughness should vanish. This objection is met if it is assumed that /(mi/zf')=a+B/V m, so that if a is a constant mi/112 tends to the limit a when m increases. A very careful discussion of the results of gaugings shows that they can be expressed more satisfactorily by this new formula than by Ganguillet and Kutter's. Putting the equation in the form (112/2gmi, § '=0'OO2594(I-f-'y/J m), where ~, - has the following values: I. Very smooth sides, cement, planed plank, 7 = 0-109 ll. Smooth sides, planks, brickwork .... 0-290 lll. Rubble masonry sides .... . 0-833 IV. Sides of very smooth earth, or pitching I '539 V. Canals in earth in ordinary condition 2-353 VI. Canals in earth exceptionally rough . -168 3

§ IOI. The Vertical Velocity Curve.-If at each point vertical

along a

representing the depth of a stream, the velocity at that point is plotted horizontally, the curve obtained is the vertical velocity curve and it has been shown by many observations that it approximates to a parabola with horizontal axis. The vertex of the parabola is at the level of the greatest velocity. Thus in fig, 104 OA is the vertical at which velocities are observed; v., is the surface; 1/, the maximum and 'lld the bottom velocity. B C D is the vertical velocity curve which corresponds with a parabola having its vertex at C. The mean velocity at the vertical is vm = élzzfz-I-21,1-I-(cl, /zl)(v.. -vd)] The Horizontal Velocity Curve.-Similarly if at each point along a horizontal representing the width of the stream the velocities are plotted, a curve is obtained called the horizontal velocity curve. In streams of symmetrical section this is a curve symmetrical about the centre line of the stream. The velocity varies little near the centre of the stream, but very rapidly near the banks. In unsymmetrical sections the greatest

velocity is at the point where the

stream is deepest, and the general

form of the horizontal velocity curve

is roughly similar to the section of

the stream.

§ 102. Curves or Contours o Equal

Velocity.-If velocities are 0 served

at a number of points at different

widths and depths in a stream, it is

possible to draw curves on the cross

section through points at which the

velocity is the same. These represent

contours of a solid, the volume

of which is the discharge of the

stream per second. Fig. 105 shows

the vertical and horizontal velocity curves and the contours of equal velocity in a rectangular channel, from one of Bazin's gaugmgs.

§ 103. Experimental Observations on the Vertical Velocity Curve- preliminary difficulty arises in observing the velocity at a given point in a stream because the velocity rapidly varies, the motion not being strictly steady. If an average of several velocities at the same point is taken, or the average velocity for a sensible period of time, this average is found to be constant. It may be inferred that A B

- Q

I

l

d

I

l

FIG. 104.

a

— 'gs-~-

5 I

tg "inn: E.”

i'§ "'§ .'.

I Y ' fc | 'ft

/1 Q, ' "L'~-~-;... fx;

|, - I .~-.g l

' ' . 1.

LT- fd 1 -

- ', | .»-~: . " ." I 1'§ ;, L ...| 1:::' ': 3 ¢ 5 3: l I V .

Vertical Velocity 5 I Hoiiizontal Velocity Cuqves; 1 Vertical Velocity Curves f:I E I Q Q 1 f 5 Cllfves

~ .. . . .,

fx Q[.1'.1°, '.°.'- -'-T '= ' = ' '“=' Q5-~'-°-4°-i';';'i*. 1' ui - Q i Q I ' Qb /I i ' ',

. ~, ; j - —;+—.4—, A-1,

(x;§ :':.;1.'.—Qi -g- - Q"-;s<-'::':;, j§ / ~ ~ ' l ' "L, ,: /

}—§ -L-~s...é..? . . t. k 3, Z I m

Contours of Egual Velocity

FIG. 105.

though the velocity at a point fluctuates about a mean value, the fluctuations being due to eddying motions superposed on the general motion of the stream, yet these fluctuations produce effects which disappear in the mean of a series of observations and, in calculating the volume of flow, may be disregarded. In the next place it is found that in most of the best observations on the velocity in streams, the greatest velocity at any vertical is found not at the surface but at some distance below it. In various river gaugings the depth d, at the centre of the stream has been found to vary from O to O-3d.

§ 104. Influence of the Wind:-In the experiments on the Mississippi the vertical velocity curve in calm weather was found to agree fairly with a parabola, the greatest velocity being at qagths of the depth of the stream from the surface. With a wind blowing down stream the surface velocity is increased, and the axis of the parabola approaches the surface. On the contrary, with a wind blowing up stream the surface velocity is diminished, and the axis of the parabola is lowered, sometimes to half the depth of the stream. The American observers drew from their observations the conclusion that there was an energetic retarding action at the surface of a stream like that due to the bottom and sides. If there were such a retarding action the position of the filament of maximum velocity below the surface would be explained.

It is not difficult to understand that a wind acting on surface ripples or waves should accelerate or retard the surface motion of the stream, and the Mississippi results may be accepted so far as showing that the surface velocity of a stream is variable when the mean velocity of the stream is constant. Hence observations of surface velocity by floats or otherwise should only be made in very calm weather. But it is very difficult to suppose that, in still air, there is a resistance at the free surface of the stream at all analogous to that at the sides and bottom. Further, in very careful experiments, P. P. Boileau found the maximum velocity, though raised a little above its position for calm weather, still at a considerable distance below the surface, even when the wind was blowing down stream with a velocity greater than that of the stream, and when the action of the air must have been an accelerating and not a retarding action. A much more probable explanation of the diminution of the velocity at and near the free surface is that portions of water, with a diminished velocity from retardation by the sides or bottom, are thrown off in eddying masses and mingle with the rest of the stream. These eddying masses modify the velocity in all parts of the stream, but have their greatest influence at the free surface. Reaching the free surface they spread out and remain there, mingling with the water at that level and diminishing the velocity which would otherwise be found there.

Influence of the Wind on the Depth at which the Maximum Velocity is found.—In the gaugings of the Mississippi the vertical velocity curve was found to agree well with a parabola having a horizontal axis at some distance below the water surface, the ordinate of the parabola at the axis being the maximum velocity of the section. During the gaugings the force of the wind was registered on a scale ranging from 0 for a calm to 10 for a hurricane. Arranging the velocity curves in three SGKS-"<I) with the wind blowing up stream, (2) with the wind blowing down stream, (3) calm or wind blowing across stream-it was found that an up-stream wind lowered, and a down-stream wind raised, the axis of the parabolic velocity curve. In calm weather the axis was at Qgths of the total depth from the surface for all conditions of the stream. Let h' be the depth of the axis of the parabola, m the hydraulic mean depth, f the number expressing the force of the wind, which may range from-l-10 to-IO, positive if the wind is up stream, negative if it is down stream. Then Humphreys and Abbot find their results agree with the expression h'/m=o-31710-06f.

Fig. }06 shows the parabolic velocity curves according to the American observers for calm weather, and for an up- or down-stream wind of a force represented by 4.

FIG. 106.

It is impossible at present to give a theoretical rule for the vertical velocity curve, but in very many gaugings it has been found that a parabola with horizontal axis fits the observed results fairly well. The mean velocity on any vertical in a stream varies from 0-85 to 0-92 of the surface velocity at that vertical, and on the average if 'vo is the surface and 'om the mean velocity at a vertical vm =-Qoo, a result useful in float gauging. On any vertical there is a point at which the velocity is equal to the mean velocity, and if this point were known it would be useful in gauging. Humphreys and Abbot in the Mississippi found the mean velocity at 0-66 of the depth; G. H. L. Hagen and H. Heinemann at 0-56 to 0-58 of the depth. The mean of observations by various observers gave the mean velocity at from 0~587 to 0'62 of the depth, the average of all being almost exactly 0-6 of the depth. The mid-depth velocity is therefore nearly equal to, but a little greater than, the mean velocity on a vertical. If amd is the mid-depth velocity, then on the average vm =0-98'o, ,, , g. § IOS. Mean Velocity on a Vertical from Two Velocity Observations. —A. ]. C. Cunningham, in gaugings on the Ganges canal, found the following useful results. Let vo be the surface, vm the mean, and was the velocity at the depth xd; then vm = %(7)o'l'37/2/lid)

= %(v~2l1d'l'U-789d>.

§ 106. Ratio of Mean to Greatest Surface Velocity, for the 'whole Cross Section in Trapezoidal Channels.-It is often very important to be able to deduce the mean velocity, and thence the discharge, from observation of the greatest surface velocity. The simplest method of gauging small streams and channels is to observe the greatest surface velocity by floats, and thence to deduce the mean velocity. In general in streams of fairly regular section the mean velocity for the whole section varies from 0-7 to 0-85 of the greatest surface velocity. For channels not widely differing from those experimented on by Bazin, the expression obtained by him for the ratio of surface to mean velocity may be relied on as at least a good approximation to the truth. Let U., be the greatest surface velocity, vm the mean velocity of the stream. Then, according to Bazin, v, ,, =v, ,—2 - /(mi).

But -vm =c/

where c is a coefficient, the values of which have been already given in the table in § 98. Hence

vm = cv»/(C+25~4)

Values of Coejicient c/(c-1-25-4) in the Formula vm =cv, /(c+25-4)

Hydraulic Very Smooth Rough Very Rough Channels Mean Depth Smooth Channels. Channels. Channels. encumbered = Channels. Ashlar or Rubble Canals in with m' l Cement. Y Brickwork. Masonry. Earth. Detritus. V 0'25 '83 '79 '69 '51 '42 0-5 ~84 ~S1 -74 ~s8 -50 075 -84 -82 ~76 -63 -55 1-0 -85 . -77 -65 -58 2~o »83 ~79 ~71 -64 3~o -80 -73 -67 4-0 -81 -75 -70 5-o ~76 ~71 6~0 '84 -77 ~72 7~o -78 -73 8-0 9~o -82 -74 I0'0 . . Is-0 -~ -79 '75 20-o -80 -76 30-0 -82 -77 40-0 . 50-0 00, 5 '79 § 107. River Bends.-In rivers flowing in alluvial plains, the windings which already exist tend to increase in curvature by the scouring away of material from the outer bank and the deposition of detritus along the inner bank. The sinuosities sometimes increase till a loop is formed with only a narrow strip of land between the two encroaching branches of the river. Finally a “ cut off ” may occur, a waterway being opened through the strip of land and the loop left separated from the stream, forming a horseshoe shaped lagoon or marsh. Professor James Thomson pointed out (Proc. Roy. Soc., 1877, p. 356; Proc. Inst. of Mech. Eng., 1879, p. 456) that the usual supposition is that the water tending to go forwards in a straight line rushes

  against the outer bank

" ' ' and scours it, at the same time creating deposits at the inner bank. »»»—> Z ian |ll.'llIllll'Nllfll' lu ig., - 11/1, /7 I - 111 111 1. muw .r1 u|u 1 A 5 Bisi § +- <<<<<<< 2 s That View is very far from a complete account of the matter, and Profill.

l l ll i' .   ll illli fll"l'il|Hl F]|/(4

1 . 1, ]/MW

A U /Z °"~° 'Z C 0 "; ll i nl i ' I le '* é 1 M ” 5

  • .

- ll I -I il "1lll Fig. 107. fessor T homson gave a much more ingenious account of the action at the bend, which he completely confirmed by experiment. When water moves round a circular curve under the action of gravity only, it takes a motion like that in a free vortex. Its velocity is greater parallel to the axis of the stream at the inner than at the outer side of the bend. Hence the scouring at the outer side and the deposit at the inner side of the bend are not due to mere difference of velocity of How in the general direction of the stream; but, in virtue of the centrifugal force, the water passing round the bend presses outwards, and the free surface in a radial cross section has a slope from the inner side upwards to the outer side (hg. 108). in curved paths, this to transverse motion. But the water im-For the greater part of the water flowing difference of pressure produces no tendency InncrBank 0uterBank mediately in contact with the rough bot"*.1 - ..@ »..»-» as-sf i t torn and sides of the "Ea., l;§ '” » » ..h'" channel is retarded, "'”” j g? and its centrifugal ?~.t;@ 7-7'*“ f~§ °°~ force is insufficient to <4?f"“"?°' -» <f-'i"°"- balance the pressure Section at M N FIG. 108. due to the greater depth at the outside of the bend. It therefore flows inwards towards the inner side of the bend, carrying with it detritus which is deposited at the inner bank. Con-Jointly with this flow inwards along the bottom and sides, the | I general mass of water must flow outwards to take its place. Fig. 107 shows the directions of flow as observed in a small artificial stream, by means of light seeds and specks of aniline dye. The lines CC show the directions of How immediately in contact with the sides and bottom. The dotted line AB shows the direction of motion of floating particles on the surface of the stream. § 108. Discharge of a River when flowing at different Depths.-When frequent observations must be made on the How of a river or canal, the depth of which varies at different times, it is very convenient to have to observe the depth only. A formula can be established giving the How in terms of the depth. Let Q be the discharge in cubic feet per second; H the depth of the river in some straight and uniform part. Then Q=aH-l-bH2, where the constants a and b must be found by preliminary gaugings in different conditions of the river. M. C. Moquerey found for part of the upper Saone, Q=64~7H-l-8-2H2 in metric measures, or Q=696H +26-SH2 in English measures. § 109. Forms of Section of Channels.-The simplest form of section for channels is the semicircular or nearly semicircular channel (fig. 109), a form now often adopted from the facility with which it can be -. . 1 j /.577 W... / // A P rtlanl Cm cnt A 3 Drzum ll? Hpe

l T ' / / / /4/ /

-fs ~=:, -<5f'= . » - .s iff. P'°S*"'f Q:QE 5.1 ~»e if jaéggg- — T 7 ~~ /f ~ <f 'ftif- rE»:f5;%3 .P f i*l, I:t—f:/- ~~ 1- ~1 - - - 1-77 "1 "4" fi 4; li ° ' », , ;, ;1, -» a> s ~ | H., Zia' #V — }- -» ...w if 4" a. ' I : -sf;-vim -=.:., »°:, -1:-" ~ gm 1, To/f9W~ a ' b 1, -611 no "'° W//, ., t ~ l"€1§ ¢, -§ '.Qaf1t”F sm. ' 1 ii 'ng/ e » 1.1 FIG. 109. executed in concrete. It has the advantage that the rubbing surface is less in proportion to the area than in any other form. Wooden channels or flumes, of which there are examples on a large scale in America, are rectangular in section, and the same form is adopted for wrought and cast-iron aqueducts. Channels built with brickwork or masonry may be also rectangular, but they are often trapezoidal, and are always so if the sides are pitched with masonry laid dry. In a trapezoidal channel, let b (fig. 110)

-~~—~ —:—71<=:%£~°@ — - —..~—1T if" fli' r/rl;-;ii1;' T:f: -; 2;:~ / V1.7 if; 'f 1T. T';"PI 'i';'§ T'€° gf' “;.°. Q 9 Q ~¢ ~ .2, o r “ 1 ~ A ~ ' A' II .Xl - . .. A F54 3 s E. -..»... . b .§ ... . g "lf, " lll l ll n Ill-llll|v | h'IQ

¥:;51-;;j~'ZE'1 = If '..'J§ jeg '9!g;;;4- .;f 1

1 /' ' /"”7'Q”e f. L 1 1: /Y /Concrcbe§ (4;/gi) FIG. IIO. be the bottom breadth, bl, the top breadth, d the depth, and let the slope of the sides be n horizontal to I vertical. Then the area of section is Sl=(b-I-nd)d=(b0-nd)d, and the wetted perimeter x=b+2d~/(n”-1-1). When a channel is simply excavated III earth it is always originally trapezoidal, though it becomes more or less rounded in course of time. The slope of the sides then depends on the stability of the earth, a slope of 2 to 1 being the one most commonly adopted. Figs. III, 112 show the form of canals excavated in earth, the former being the section of a navigation canal and the latter the section of an irrigation canal. § 110. Channels of Circular Section.-The following short table facilitates calculations of the discharge with different depths of water in the channel. Let T be the radius of the channel section; then for a depth of water=:<r, the hydraulic mean radius is yr and the area. of section of the waterway vrz, where K, u, and v haveethe following values:- Depth of water in terms of radius Hydraulic mean depth) Waterway in terms oi(= square of radius cms() our 0598 .1067 .1651 .228 .294 .370

Q K = .01 .05 in termsofradius § lJ~= .oo668 .O32I .0523 .OQ63 .1278 .1574 .1852 .2142 V . . > 5 | 5 40 .45 .50 .55 .60 .65 .70 .75 .80 .85 .90 .95 1.0 242 .269 .293 .320 .343 .365 .3871.4o8 .429 .449 .466 .484 .Soo 450 .532 .614 .709 .795,885 .979 1.075 1.175 1.276 1.371 1.470 1.571 lnBan.lc

§ III. Egg-Shaped Channels or Sewers.-In sewers for discharging storm water and house drainage the volume of flow is extremely variable; and there is a great liability for deposits to be left when the How is small, which are not removed during the short periods when the flow is large. The sewer in consequence becomes choked. could be found satisfying the foregoing conditions. To render the problem determinate, let it be remembered that, since for a given discharge Qoo Q/X, other things being the same, the amount of excavation will be least for that channelgwhich has the least wetted perimeter. Let d be the depth and b the bottom width of the channel, and let the sides slope n horizontal to I vertical 1

Di Cafting

l

(fig. 114), then

I 1' ~, . I, i /1 » 1 ~ 1 I -H, Q=(b+nd)d. u-s-14~9- -Q ~a~4<——33~|o—-~»-4<<s'-fo-»x-~—21~4—>, “ 1 1 28 1: 1 | f b x=b-l-2d/ (nz--1). 1 1 1 l l Wy, Both Sl and X are to be minima. 1 ' i i i l 'ww Differentiating, and equating to l' 1 1 I *W , ,, ... h a . Ze;-0 1 ~

ff Sf af 14: (db/dd +n>d +1 +114 = 0, 4455* Ein" 'gag ~s- /" “" " db/dd+2~/ (n2+1)=0; 1s ., —':=, ,1..;, ¢, .Wm

~1n'u1W*§§ })§ M eliminating db /dd, 1 ' ' 1 ' N pi-24 (n2+i>}d+b+nd=0rr b=2{, /<n2+1)-md.

FIG. III.*-SC3lé 20 ft. = 1 in. But S2/X=(b+hd)d/{b+2d/(n2+1)}.

I Inserting the value of b,

-, , . .1 . .. . .:

L ... .,20'.0!;-1: — -1 aa-o»~=m>g+~—— ~ 14-u-o—<~—f Slowking4Farmbrough's revenge»>?1f1+2a-oq<-»- — —120~o1——~~—»§ 1"=9/X=l2d/ ( 2+2I)'~"lll/ 1 2 1 5 = L, . .. 10p:.02— ... 4 5 = ' = Th wig/ (M +1>-§ 11dll=§ d, “ , ,, ,, ,, ,, , e tem ~, - -:.. .— — — —, - — -, -... ~ —1. ~¢ ¢, W-»-»-f-51553, ..-., .», ., . ' 1 ~ ' ~ - -§ (§ ®>°'lll|lllll“' Tf§ @)§§ "; § J=r- - "”l@%3§ W5'f"""*§ $77'?.'@:, § ¥s;» m U} - lMm~' vr at ls, Wlt given 51 e 5 QPeS» " ” ""' '”'“""" — - 4 ”?'>». §§ g~, , 1 ' W —~~ T § <~ % " at ' ' / I' ' the section is least for a given ~f" ~wf;° ' ;/V "

FIG. 112.—Scale 80 ft. =1 in.

To obtain uniform scouring action, the velocity of flow should be constant or nearly so; a complete uniformity of velocity cannot be obtained with any form of section suitable for sewers, but an approximation to uniform velocity is obtained by making the sewers of oval section. Various forms of oval have been suggested, the simplest being one in

which the radius of the

crown is double the radius

4- —~—- 4- —— ~o

discharge when the hydraulic mean depth is half the actual depth. A simple construction gives the form of the channel which fulfils this condition, for it can be shown that when m=§ <l the sides of the channel are tangential to a semicircle drawn on the water line.

I Since SZ/X = ed,

1 therefore S2 = Xd. (I)

Let ABCD be the channel (fig. 115); from E the centre of AD drop perpendiculars EF, EG, EH on the sides. Z E of the invert, and the i Let "“'§ '1" ' greatest width is two- AB=CD=a; BC=b; EF=EH=c; and EG=d. g » thirds the height. The 9=ar@a AEB~l-BEC-l-CED I ' } section of such a sewerl =aC+lbd 1 I ' is shown in fig. II3, the 1 x=2a+2 5 "2~~.> Y numbers marked on the § ' "'l" """°' " Z) /»' figure being proportional Pllttlflg fh€S€ V€1111€S 111 (I), ¥"' 6 I/" Problems on ac-I-%bd= (a+%b)d; and hence c=d. "~ ' ' ~~ hannels in which the T § "~ :;;><;f, Flow is Steady and at ...K .., . D 1 <»" '~, »' Uniform Velocity.-The 5 A 2 " /, /'x general equations given E '. "f7< 1 in §§ 96, 98 are H g, '4"' ' 5 r=<=<1+/S/m>. <1> § e “ ' ' ' tv-/2g =m1: (2) E

FIG- 113- Q=Qp (3) | 1 .-Problem I.-Given the transverse section of stream and dis- B G C charge, to find the slope. From the dimensions of the section FIG 115 Eng fl and m; from (1) find gf, from (3) find v, and lastly from (2) n 1.

Problem II.-Given the transverse section and slope, to find the discharge. Find v from (2), then Q from (5). Problem III.-Gi'en the discharge and slope, and either the breadth, depth, or general form of the section of the channel, to determine its remaining dimensions. This must generally be solved by approximations. A breadth or depth or both are chosen, and the discharge calculated. If this is greater than the given discharge, the dimensions are reduced and the discharge recalculated. Since m lies generally between the limits m=1l and m=§ d, where d is the depth of the st1'eam, and since, moreover, the velocity varies as xl (m) so that an error in the value of m leads only to a much less error in the value of the velocity calculated from it, we may proceed thus. Assume a value for m, and calculate '11 from it. Let v1 be this first approximation to U. Then Q/U1 is a first approximation to Q, say 91. With this value of S2 design the section of the channel; calculate a second value for Ta 'a r

fi 'av 1

1-1 4 =

T y —nd—sg. ...... (, , ., ..

FIG. 114.

trapezoidal in section (fig. 114), and m; calculate from it a second

value of 11, and from that a

second value for Sl. Repeat

the process till the successive values of m approximately

coincide.

§ 113. Problem IV. Most

Economical Form of Channel

for given Side Slopes.-Sup

pose the channel is to be

that the sides are to have a

given slope. Let the longitudinal slope of the stream be given, and also the mean velocity. An

infinite number of channels

That is, EF, EG, EH are all equal, hence a semicircle struck from E with radius equal to the depth of the stream will pass through F and H and be

tangential to the sides of

the channel.

To draw the channel,

describe a semicircle on

a' horizontal line with

radius=depth of channel.

""'" "H+ —» -6 —— >¢

FIG. 116.

The bottom will be a

horizontal tangent of that

| sizmicircle, and the sides tangents drawn at the required side s opes.

The above result may be obtained thus (fig. 116):- X=b-l-2d/sin B. (1)

Sl=d(b-Pd cot /3);

fl/d=b-l-d cot B; (2)

SZ/dz =b/d-I-cot B. (3)

From (I) and (2),

X =9/ol-d cot B-l~2d/sin B.

This will be a minimum for

dx, /dd = SZ/dz-|~cot, B-2/sin /3 =0, or SZ/d” =2 cosec. B-cot B. (4) or d =/ {S2 sin B/(2 -cos /3)}. From (3) and (4),

b/d=2(1 -cos B)/sin /S=2 tan é/3. Proportions of Channels o Maximum Discharge for given Area and Side Slopes. Depth 0 channel=d; Hydraulic mean depth=§ d; Area of section =Q.

.

- ~ ° f I Top width =

?§§ ;§ ..;§ 
§ :.°§ . me;i§ .tsa*;

ope.

Semicircle . .. .. I'57Id2 0 2d

Semi-hexagon . 6o° o' 3:5 I°732d2 I'155d 2'3I0d Semi-square . 90° o' o; 1 zd' 2d 2d 75° 58' I:4 1-812112 1 56211 2-o62d 63° 26' 1: 2 1-736:12 1-23611 2'236d 53° 8' 3:4 I~75Od2 d 2'500d

45° o' I:I I-828122 0°828d 2-828d 38° 40' Ig; 1 I'952d2 0'702d 3-202d 33° 42' 1%: I 2-IO6d2 o~6o6d 3-6o6d 29° 44' 1%: 1 2-282d2 o-532d 4~032d 26° 34' 2; 1 2'472112 O'472d 4-472d 23° 58' 2%: 1 2-674d=' 0°424d 4~924d 21° 48' 2;: I 2-885112 o-385d 5'385d 19° 58' 2%:1 3'IO4d2 o-354d 5-854d 18° 26' 3: 1 3~325d2 o-325d 6-325d Half the top width is the length of each side slope. The wetted perimeter is the sum of the top and bottom widths § 1 14. Form of Cross Section of Channel in which the Mean Velocity is Constant with Varying Discharge.-In designing waste channels from canals, and in some other cases, it is desirable that the mean velocity should be restricted within narrow limits with very different volumes of discharge. In channels of trapezoidal form the velocity increases and diminishes with the discharge. Hence when the discharge is large there is danger of erosion, and when it is small of silting or obstruction by weeds. A theoretical form of section for which the mean velocity would be constant can be found, and, although this is not very suitable for practical purposes, it can be more or less approximated to in actual channels. Let fig. 117 represent the cross section of the channel. From the symmetry of the section, only half the channel need be considered. xi

fi:;:~':;~';°-°~~° °-'° —°~- - ~~°-~°- if """" i """'°"“"

ci ll

Scale 1% Inch =1Foot.

FIG. 117.

Let obac be any section suitable for the minimum How, and let it be required to find the curve beg for the upper art of the channel so that the mean velocity shall be constant. Take o as origin of coordinates, and let de, fg be two levels of the water above ab. Let ob=b/2; de=y, fg=y-l-dy, od=x, of=x+dx; eg=ds. The condition to be satisfied is that v=c / (mi)

should be constant, whether the water-level is at ob, de, or fg. Consequently m = constant = k

for all three sections, and can be found from the section obac. Hence also

Increment of section 3Lclic k

Increment of perimeter- ds "

y2dx2 = kldsf = k'(dx2+dy2) and dx =kdy/ / (y'-k'). Integrating,

x =k logf {y-l- V (y'-k')l-l-constant; and, since y=b/2 when x=o,

x=k logs lly+ w/ (yi-k')}/l%b+ w/ (tif-k')llglssuming values for y, the values of x can be found and the curve rawn.

The figure has been drawn for a channel the minimum section of which is a half hexagon of 4 ft. depth. Hence k=2; b=9-2; the rapid flattening of the side slopes is remarkable. Srmuv MOTION or WATER IN OPEN CHANNELS or VARYING Cnoss SECTION AND SLOPE

§ 1 15. In every stream the discharge of which is constant, or may be regarded as constant for the time considered, the velocity at different places depends on the slope of the bed. Except at certain exceptional points the velocity will be greater as the slope of the bed is greater, and, as the velocity and cross section of the stream vary inversely, the section of the stream will be least where the velocity and slope are greatest. If in a stream of tolerably uniform slope an obstruction such as a weir is built, that will cause an alteration of flow similar to that of an alteration of the slope of the bed for a greater or less distance above the Weir, and the originally uniform cross section of the stream will become a varied one. In such cases it is often of much practical importance to determine the longitudinal section of the stream. The cases now considered will be those in which the changes of velocity and cross section are gradual and not abrupt, and in which the only internal work which needs to be taken into account is that due to the friction of the stream bed, as in cases of uniform motion. Further, the motion will be supposed to be steady, the mean velocity at-each given cross section remaining constant, though it Varies from section to section along the course of the stream. Let fig. 118 represent a longitudinal section of the stream, AeA, being the water surface, BOBI the stream bed. Let AOBU, AIB1 be 2 4' ffl" <'» Qs

"'~4"¢' al

T Y/'il l I . lar itil,

B0 W;/tar

B if W

FIG. 118.

cross sections normal to the direction of flow. Suppose the mass of water AUBOAIBI comes in a short time 6 to CQDOCIDI, and let the work done on the mass be equated to its change of kinetic energy during that period. Let l be the length ADA; of the portion of the stream considered, and z the fall of surface level in that distance. Let Q be the discharge of the stream per second. Change of Kinetic Energy.-At the end of the time 0 there are as many particles possessing the same velocities In the space CQDOAIB, as at the beginning. The

change of kinetic energy is

therefore the difference of

the kinetic energies of

AnB0C0Dg and A;B1CID1.

Let fig. 119 represent the

cross section AOBO, and let

w be a small element of its

area at a point where the

velocity is v. Let S20 be the

~ Wi.., - .f

We a-

ts. ... . — — - , f.g.,

- - ~~- -~ ~-'- - -*

”“ Elw ~'§ *“"

"4 .

ss 1 1

- 'Nw . <>°»gfgirs

0 Wg, - u' .

ss ;», agyféxo, La

FIG. 1 19.

whole area of the cross section and ug the mean velocity for the whole cross section. From the definition of mean velocity we have no =Ewv/90. s.

Let v=u0-I-'w, where w is the difference between the velocity at the small element w and the mean velocity. For the whole cross section, Eww = 0.

The mass of fluid passing through the element of section oi, in 0 seconds, is (G/g)wv0, and its kinetic energy is (G/2g)w1J50. For the whole section, the kinetic energy of the mass AOBOCQDO passing in 0 seconds is A

(GH/2g)Zwv3 = (G0/2g)Ew(u03+3u0”w+3u0w” -l-103), = (GH/2g){u03SZ ~l-Eww2(3u0 J, -ze) }. The factor 3u0-I-fw is equal to 2ii0-I-v, a quantity necessarily positive. Consequently ZIwz'3>S20u03, and consequently the kinetic energy of AOBQCDDU is greater than (GH/2g)S10u03 or (G0/2g)QufF.

which would be its value if all the particles passing the section had the same velocity uo. Let the kinetic energy be taken at a(G6/2g)S29ii03 = a(G0/2g)Qu02,

where a is a corrective factor, the value of which was estimated by ]. B. C. ]. Bélanger at I~I.' Its precise value is not of great importance. In a similar way we should obtain for the kinetic energy of A1B1C1D, the expression

a(G0/2g)S2, u13 = a(G6/'2g)Qu12,

where $21, ui are the section and mean velocity at AIB1, and where ui may be taken to have the same value as before without any important error.

Hence the change of kinetic energy in the whole mass A0B0A1B in 6 seconds is

a(G0, '2g)Q(n,2-nog). (I)

Motive Work of the Weight and Pressures.-Consider a small filament anal which comes in 0 seconds to cocl. The work done by gravity during that movement is the same as if the portion agco were carried to a1c1. Let dQ0 be the volume of aoco or a1c1, and yo, yi the depths of ao, al from the surface of the stream. Then the volume 1 Boussinesq has shown that this mode of determining the corrective factor a is not satisfactory. dQ6 or GdQ6 pounds falls through a vertical height z+y1- yo, and the work done by gravity is GdQ0(2+;v1-yo)-Putting pa for atmospheric pressure, the whole pressure per unit of area at ag is Gy0+p, ,, and that at at is -(Gyi-l-pa). The work of these pressures is G0/o+P»/G-yi-Pa/G)dQ0=G(;'°-;'1)dQ9-Adding this to the work of gravity, the whole work is GzdQ0; or, for the whole cross section, GzQ6. (2) Work expended in Overcoming the Friction of the Stream Bed.-Let A'B', A”B" be two cross sections at distances 5 and s-{-ds from AOBD. Between these sections the velocity may be treated as uniform, because by hypothesis the changes of velocity from section to section are gradual. Hence, to this short length of stream the equation for uniform motion is applicable. But in that case the work in overcoming the friction of the stream bed between A'B' and GQ0§ (1¢2/2g) (x/SMS, where u, X, SZ are the mean velocity, wetted perimeter, and section at l{£)B'. Hence the whole work lost in friction from A0130 to A1Bi wi e A”B" is ZGQH grae/2g><X/n>d$. <3> Equating the work given in (2) and (3) to the change of kinetic energy given in (1), ~ Y <1(GQ0/23) (W-1402) =bQ2@ ~§ 3Q9f<fs“(1¢”/2g)(x/S2)dS; 2 = a(u{* -1402)/2g+ 0§ '(u”/2g) (X/Q)ds. § 116. Fundamental Dijerential Equation of SteadyVariedMotion.-Suppose the equation just found to be applied to an indefinitely short length ds of the stream, limited by the end sections ab, albi, taken for simplicity normal to the stream bed (fig. 120). For that short length of stream the fall of surface level, or difference of level of f~;;:; -~——~ ~- —- — t

4 I 5, ' "2'~/d

lg, . . u /1|;; I r 1/ hfdh J 1 5 /%»'.. u 6fhEi""'~~, d I f *du 0 "V ' 4"~- if

/ %/ / lb/ /, -W - i@f”f< c/ FIG. 1 20. a and al, may be written dz. ul, the term (uoz-uiz)/2 Also, if we write u for ug, and u+du for g becomes udu/g. Hence the equation applicable to an indefinitely short length of the stream is dz=udu/g+(x/S2>f(u”/2g)dS- (I) From this equation some general conclusions may be arrived at as to the form of the longitudinal section of the stream, but, as the investigation is somewhat complicated, it is convenient to simplify it by restricting the conditions of the problem. Modifcation of the Formula for the Restricted Case of a Stream /iowing in a Prismatic Stream Bed of Constant Slope.-Let i be the constant slope of the bed. Draw ad parallel to the bed, and ac horizontal. Then dz is sensibly equal to a'c. The depths of the stream, h and h-l-dh, are sensibly equal to ab and a'b', and therefore dh=a'd. Also cd is equal to ids. Hence dz=a'c=cd-a'd=ids-dh. (2) Since the motion is steady the fall of the bed in the distance ds, and is Q =S't1.¢ =constant. Differentiating, SZdu+udS2=o; du = -udfl/Q. Let x be the width of the stream, then dQ=xdh very nearly. Inserting this value, » du = - (nx/St)dh. (3) Putting the values of du and dz foufid in (2) and (3) in equation (1), i¢iS°-dh = - (Mix/g5l)lil1+(x/9)$'(“2/2£)d5dh/ds = it - (x/S2)r(u*/2g>}/ l I - (142/gl (x/9)-l (4) Further Restriction to the Case of a Stream of Rectangular Section and of Indejinite Width.-The equation might be discussed in the form just given, but it becomes a little simpler if restricted in the way just stated. For, if the stream is rectangular, xh=$2, and if x is large compared with h, Q/X =xh/x =h nearly. Then equation (4) becomes dh/ds ='i(1 -(142/2gih)/(I -142/gh). (5) § 117. General Indications as to the Form of Water Surface furnished by Equation (5).-Let AOA, (fig. 121) be the water surface, BUB, the bed in a longitudinal section of the stream, and ab any section at a distance s from Bo, the depth ab being h. Suppose BOB1, BOAO taken as rectangular coordinate axes, then dh/ds is the trigonometric tangent of the angle which the surface of the stream at a makes with the axis BOB, This tangent dh/ds will be positive, if the stream is increasing in depth in the direction B0Bi; negative,

I 4

G F i A 1: 1 9 1 » v~',

B f .

0 Zi; ¢ I 4'/4 I, ' 1 {*-~~ ?/¢/ 1 f / -»?@, ¢, ,%f5 I -~.. ' 4% 1 ~-., '» & 1 / ~”%:iq e. -..- i - . f 5% . FIG. 121. if the stream is diminishing in depth from B0 towards Bl. If dh/ds =o. the surface of the stream is parallel to the bed, as in cases of uniform motion. But from equation (4) dh/ds=0, if i-(x/Qlffu'/2g)=0; j -'-s“(“2/23) = (9/>g)¢=m¢, which is the well-known general equation for uniform motion, based on the same assumptions as the equation for varied steady motion now being considered. The case of uniform motion is therefore a limiting case between two different kinds of varied motion; Consider the possible changes of value of the fraction (1 -fu*/zgih)/(I -uz/gh). As h tends towards the limit 0, and consequently u is large, the numerator tends to the limit-00 . On the other hand if h=°o, in which case u is small, the numerator becomes equal to I. For a value H of h given by the equation 1 -fu'/2giH =o, H = fu*/2gi, we fall upon the case of uniform motion. The results just stated may be tabulated thus:- For h=o, H, >H, oo, the numerator has the value -oo, o, > 0, 1. Next consider the denominator. If h becomes very small, in which case u must be very large, the denominator tends to the limit - co As h becomes very large and it consequently very small, the denominator tends to the limit 1. For h=u”/g, or u=x/ (gh), the denominator becomes zero. Hence, tabulating these results as before:- For h=o, 142/g, > uf/g, oo, the denominator becomes -oo, 0, > 0, I. § 118. Case 1.-Suppose h> 142/g, and also h> H, or the depth greater than that corresponding to uniform motion. In this case dh/ds is positive, and the stream increases in depth in the direction of flow. In fig. 122 let BOB1 be the bed, CGC, a line parallel to the bed and at a height above it equal to H. By hypothesis, the surface Ao Co fl IZA -» —- ..

H 1; D 5 """D

0, 1 ""'

VN(C1 -..fs- >ssg

FIG. 122. AQA1 of the stream is above CQC1, and it has just been shown that the depth of the stream increases from BD towards Bl. But going up stream h approaches more and more nearly the value H, and therefore dh/ds approaches the limit 0, or the surface of the stream is asymptotic to COC1. Going down stream h increases and u diminishes, thenumeratorand denominator of the fraction(1 - § "lL2/2gth)/ (1 -uf/gh) both tend towards the limit 1, and dh/ds to the limit i. That is, the surface of the stream tends to become asymptotic to a horizontal line DUD1. The form of water surface here discussed is produced when the flow of a stream originally uniform is altered by the construction of a Weir. The raising of the water surface above the level CaCl is termed the backwater due to the weir. § 119. Case 2.-Suppose h>u2/g, and also h<H. Then dh/ds is negative, and the stream is diminishing in depth in the direction of flow. In fig. 123 let BOB, be the stream bed as before; CDC, a line drawn parallel to B0B1 at a height above it equal to H. By hypothesis the surface AQA1 of the stream is below CUC1, and the depth has just been shown to

Cf; diminish from Bu

A towards Bl. Going

0, up stream 11 ap-4

I, Q proaches the limit

f J! T H, and dh/ds tends

go 4' to the limit' zero.

A ' . That is, u stream

/M / if VB AOA, is ag/mptotic

1' to COC1. Going down

I, -IG 123 ' stream h diminishes

and u increases; the

inequality h>u2/g diminishes; the denominator of the fraction (1I-(15/zcgilz)/(1-1¢2/'g}r) tends to the limit zero, and cons ”

equent y 1, 5 tends to oo . That is, down stream AOA, tends t o a direction perpendicular to the bed. Before, however, this limit was reached the assumptions on which the general equation is based would cease to be even approximately true, and the equation would cease to be applicable. The filaments would have a relative motion. which would make the influence of internal friction in the fluid too important to be neglected. A stream surface of this form may be produced

if there

is an abrupt

fall in the bed

of the stream

(fig. 124).

On the Ganges

canal, as originally

con-

structed, there

w e r e a b r u p t

falls precisely

of this kind,

and it appears

that the lowering of the water surface and increase of velocity which such falls occasion, for a distance of some miles up stream, nas not foreseen. The result was that, the velocity above the falls being greater than was intended, the bed was scoured and l

consir erabe damage was done to the works. “ ¥Vhen the canal was first opened the water was allowed to pass freely over the crests of the over falls, which were laid on the level of the bed of the earthen channel; erosion of bed and sides for some miles stream is of indefinitely great width, so that m =H, then § 'u2/2g =iH, and H = fu”/2gi. Consequently the condition stated above involves that

5142/2gi<u2/g, or that f> 5°/2.

If such a stream is interfered with by the construction of a weir which raises its level, so that its depth at the weir becomes h1> u2, 'g, then for a portion of the stream the depth h will satisfy the conditions h <u2/g and h> H, which are not the same as those assumed in the two previous cases, At some point of the stream above the weir the depth h becomes equal to 142/g, and at that point dh/ds becomes infinite, or the surface of the stream is normal to the bed. It is obvious that at that point the influence of internal friction will be too great to be neglected, and the general equation will cease to represent the true conditions of the motion of the water. It is known that, in cases such as this, there occurs an abrupt rise of the free surface of the stream, or a standing wave is formed, the conditions of motion in which will be examined presently. It appears that the condition necessary to give rise to a standing wave is that i> § '/2. Now § ' depends for different channels on the roughness of the channel and its hydraulic mean depth. Bazin calculated the values of § ' for channels of different degrees of roughness and different depths given in the following table, and the corresponding minimum values of i for which the exceptional case of the production of a standing wave may occur. Slope below

which a Stand-Nature

of Bed of Stream. ing Vlfave is

impossible in

Standing Wave Formed,

Slope in feet Least Depth

feet per foot. per foot. in feet.

0-002 0-262

Very smooth cemented surface 0'00147 0-003 -098 0-004 -065

0'003 '394

Ashlar or brickwork 0-00186 0-004 -197 0-006 -098

0-004 ' 1-181

Rubble masonry 0'OO235 0-006 -525 0-010 ~262

0-006 3-478

Earth 0-00275 0-010 I'542

0-015 -919

must

ingly

up rapidly followed, and 1t soon became apparent that means be adopted for raising the surface of the stream at those points (that is, the crests of the falls). Planks were accord fixed in th - 'e

grooves above the bridge arches, or temporary weirs were formed over which the water was allowed to fall; in some cases the surface of the water was thus raised above its normal height, causing a backwater in the channel above" (Cr0fton's Report on the Ganges Canal, p. 14). Fig. 125 represents in an exaggerated form what probably occurred, the diagram being intended Q-.e

STANDING WAVES

§ 121. The formation of a standing wave was first observed by Bidone. Into a small rectangular masonry channel, having a slope of O°O23 ft. per foot, he admitted water till it flowed uniformly with a depth of O'2 ft. He then placed a plank across the stream which raised the level just above the obstruction to 0-95 ft. He found that the stream above the obstruction was sensibly unaffected up to a point 15 ft. from it. At that point the depth suddenly increased from 0-2 ft. to 0-56 ft. The velocity of the stream in the part unaffected by the obstruction was 5-54 ft. per second. Above the point where the abrupt change of depth occurred u2=5-542=30-7, and gh=32-2 X0-2=6-44; hence u2 was>gh. ]ust below the abrupt change of depth u=5-54><o-2/0-56=1-97; u2=3-88; and gh-32'2X0'56= I8~03; hence at this point u2<gh. Between these two B ~ " points, therefore, u2=gh; and the condition for the production of a ""~- s C standing wave occurred.

"'r~-' ' /1 fl Theghange of level at a staardilig wave may be found thgs. bLe§ 1 17, ~~ g. 12 represent the ongitu ma section 0 a stream an a, c %4/%%y%y, ,-

2, if B

— /// /. / I//./4/y /, ~, fy 2/ . Q "H i -aflit

= = = J 2 z>>>»»-

5 5 5 5 5 5

/ "'0 Eb gb, 5 di

Fla 125

FIG. 126.

to represent some miles' length of the canal bed above the fall. AA parallel to the canal bed is the level corresponding to uniform motion with the intended velocity of the canal. In consequence of the presence of the ogee fall, however, the water surface would take some such form as BB, corresponding to Case 2 above, and the velocity would be greater than the intended velocity, nearly in the inverse ratio of the actual to the intended depth. By constructing a weir on the crest of the fall, as shown by dotted lines, a new water surface CC corresponding to Case I would be produced, and by suitably choosing the height of the weir this might be made to agree approximately with the intended level AA. § 120. Case 3.-Suppose a stream flowing uniformly with a depth h<u'/g. For a stream in uniform motion fu'/2g=mi, or if the cross sections normal to the bed, which for the short distance considered may be assumed horizontal. Suppose the mass of water abcd to come to a'b'c'd' in a short time t; and let ug, ul be the velocities at ab and 611,520,521 the areas of the cross sections. The force causing change of momentum in the mass abcd estimated horizontally is simply the difference of the pressures on ab and cd. Putting ho, hi for the depths of the centres of gravity of ab and cd measured down from the free water surface, the force is G(h0S74;-hlfll) pounds, and the impulse in t seconds is G (hnflo-hlfll) I second pounds. The horizontal change of momentum is the difference of the mornenta of cdc'd' and aba'b'; that is,

(G/gl (91“1”-9o“0”)¢ Hence, equating impulse and change of momentum, G(h090"'h1H1ll=(G/§)(Qi'l»t12-Q0'lto2)f} -'- ho9o“hifli = (§ 111412-901402)/5 (I) For simplicity let the section be rectangular, of breadth B and depths H0 and Hi, at the two cross sections considered; then ho = il-Io, and h, = § H1. Hence

H02-H12=(2/g)(Hiu12-Houoz).

But, since Q9u0=$Z1u1, we have

1412 = 1¢o2Ho2/H121

I'102'-'I'l12: (2u02/lg)(H02/H1”H0). (2) This equation is satisfied if H0=I-ll, which corresponds to the case of uniform motion. Dividing by H0-Hr, the equation becomes (HI/Ho)(H0'l'H1)=21402/.QI (3)

H1= V (2u4J2H0/g+ il-ltr) —§ H0. (4) In Bid0ne's experiment u1)=5-54, and H0=0'2. Hence H1=0-52, which agrees very well with the observed height. § 122. A standing wave is frequently produced at the foot of a weir. Thus in the ogee falls originally constructed on the Ganges canal a standing wave was observed as shown in fig. 127. The water falling over the weir crest A acquired a very high velocity on the A

steep slope AB, and the section of the stream at B became very small. It easily happened, therefore, that at B the depth h<u”/g. In flowing along the rough apron of the weir the velocity 14 diminished and the depth h increased At a point C, where h became equal to ui'/g, the conditions for producing the standing wave occurred. Beyond C the free surface abruptly rose to the level corresponding to uniform motion with the assigned slope of the lower reach of the canal.

A standing wave is sometimes formed on the down stream side of bridges the piers of which obstruct the flow of the water. Some interesting cases of this kind are described in a paper on the “ Floods in the Nerbudda Valley” in the Proc. Inst. Civ. Eng. vol. xxvii. p. 222, by A. C. Howden. Fig. 128 is compiled from the data given in that paper. It represents the section of the stream at pier 8 of -..—.... . -.. ..

-, .. | &

-

$1>

A T

- e

... ., ...

Q, 'i"':, '1'.

Z r' ', -»cl

1 Z

° .

QI*

I ' Q

N I

7 4. /, /, VH 1 rf/p /7

/Z /% »%%%//M%W lM%%%W

FIG. 128.

the Towah Viaduct,

during the flood of 1865.

The ground level is not

exactly given by Howden,

but has been inferred

from data given

o nan other drawing. The

velocity of the stream

was not observed, but

the author states it was

probably the same as at

the Gunjal river during

a similar flood, that is

16~58 ft. per second.

Now, taking the depth

on the down stream face

of the pier at 26 ft., the

velocity necessary for the

production of a standing

wave would be u= / (gh)

=/(32-2><26)=29 ft.

per second nearly. But

the velocity at this

point was probably from Howden's statements I6-58><§ 2=25-5 ft., an agreement as close as the approximate character of the data would lead us to expect.

XI. ON STREAMS AND RIVERS

§ 123. Catchment Basin.-A stream or river is the channel for the discharge of the available rainfall of a district, termed its catchment basin. The catchment basin is surrounded by a ridge or watershed line, continuous except at the point where the river finds an outlet. The area of the catchment basin may be determined from a suitable contoured map on a scale of at least 1 in 100,000. Of the whole rainfall on the catchment basin, a part only finds its way to the stream. Part is directly re-evaporated, part is absorbed by vegetation, part may escape by percolation into neighbouring districts. The following table gives the relation of the average stream discharge to the average rainfall on the catchment basin (Tiefenbacher). Ratio of average Loss by Evaporation? Discharge to &c., in per cent of

average Rainfall. total Rainfall. Cultivated land and spring-forming declivities °3 to '33 67 to 70

/Vooded hilly slopes . . -35 to -45 55 to 65 Naked unfissured mountains 1 -55 to -60 40 to 45 § 124. Flood Discharge.—The flood discharge can generally only be determ-ined by examining the greatest height to which floods have been known to rise. To produce a flood the rainfall must be heavy and widely distributed, and to produce a flood 'of exceptional height the duration of the rainfall must be so great that the flood waters of the most distant affluents reach the point considered, simultaneously with those from nearer points. The larger the catchment basin the less probable is it that all the conditions tending to produce a maximum discharge should simultaneously occur. Further, lakes and the river bed itself act as storage reservoirs during the rise of water level and diminish the rate of discharge, or serve as flood moderators. The influence of these is often important, because very heavy rain storms are in most countries of comparatively short duration. Tiefenbacher gives the following estimate of the flood discharge of streams in Europe:-

Flood discharge of Streams

per Second per Square l/lile

of Catchment Basin.

In flat country ...... 8-7 to I2'5 cub. ft. In hilly districts ...... 17-5 to 22-5, , In moderately mountainous districts 36-2 to 45-0, , In very mountainous districts 50-0 to 75-0, , It has been attempted to express the decrease of the rate of flood discharge with the increase of extent of the catchment basin by empirical formulae. Thus Colonel P. P. L. O'Connell proposed the formula y=Mx/ x, where M is a constant called the modulus of the river, the value of which depends on the amount of rainfall, the physical characters of the basin, and the extent to which the floods are moderated by Storage of the water. If M is small for any given river, it shows that the rainfall is small, or that the permeability or slope of the sides of the valley is such that the water does not drain rapidly to the river, or that lakes and river bed moderate the rise of the floods. If values of M are known for a number of rivers, they may be used in inferring the probable discharge of other similar rivers. For British rivers M varies from 0-43 for a small stream draining meadow land to 37 for the Tyne. Generally it is about 15 or 20. For large European rivers M varies from 16 for the Seine to 67-5 for the Danube. For the Nile M = I I, a low value which results from the immense length of the Nile throughout which it receives no affluent, and probably also from the influence of lakes. For different tributaries of the Mississippi M varies from 13 to 56. For various Indian rivers it varies from 40 to 303, this variation being due to the great variations of rainfall, slope and character of Indian rivers. In some of the tank projects in lndia, the flood discharge has been calculated from the formula D =C{'/112, where D is the discharge in cubic yards per hour from 11 square miles of basin. The constant C was taken =6I,523 in the designs for the Ekrooka tank, =75,000 on Ganges and Godavery works, and = 10,000 on Madras works. § 125. Action af a Stream on itsBed.-If the velocity of a stream exceeds a certain limit, depending on its size, and on the size, heaviness, form and coherence of the

material of which its bed is composed, it scours its bed and

carries forward the materials.

The quantity of material which

a given stream can carry in

suspension depends on the size

and density of the particles in

suspension, and is greater as

the velocity of the stream is greater. If in one part of its course the velocity of a stream is great enough to scour the bed and the water becomes loaded with silt, and in a subsequent part of the river's course the velocity is diminished, then part of the transported material must be deposited. Probably deposit and scour go on simultaneously over the whole river bed, but in some parts the rate of scour is in excess of

the rate of deposit, and

- "

1

f

X *


"7"'W%;{'7*'WW7'/ Z//W //¢%€/ /< FIG. 1 29.

in other parts the rate ' A

of deposit is in excess

of the rate of scour. , ';

Deep streams appear to "' "

have the greatest scour- //' '

v / '

ilgopgxer ing/O§ ;hi';

that the difference is

strictly a difference of

transporting, not of scouring action. Let fig. 129 represent a section of a stream. The material lifted at a will be diflused through the mass of the stream and deposited at different distances down stream. The average path of a particle lifted at a will be some such curve as abc, and the average distance of transport each time a particle is lifted Fic.. 130. will be represented by ac. In a deeper stream such as that in fig. 130, the average height to which particles are lifted, and, since the rate of vertical fall through the water may be assumed the same as before, the average distance a'c' of transport will be greater. Consequently, although the scouring action may be identical in the two streams, the velocity of transport of material down stream is greater as the depth of the stream is greater. The efifect is that the deep stream excavates its bed more rapidly than the shallow stream. § 126. Bottom Velocity at which Scour coinrnencesf-The following bottom velocities were determined by P. L. G. Dubuat to be the maximum velocities consistent with stability of the stream bed for different materials. Darcy and Bazin give, for the relation of the mean velocity vm and bottom velocity rfb. ~v, ,, =z~1, +1o-87/ (mi). But ~/ mi ='l'm/ (I/2g): vm=vb/(1-10-874 (K/2g>)-Taking a mean value for § ', we get 1;, ,, = 1 -312215, and from this the following values of the mean velocity are obtained:- Bottom Velocity Mean Velocity =vb. =1, ', ,, . 1. Soft earth o-25 -33 2. Loam . 0-50 -65 3. Sand . . l'00 1-30 4. Gravel 2-00 2-62 1 5. Pebbles .... 3-40 4-46 6. Broken stone, flint 4-O0 5-25 7. Chalk, soft shale 5-00 6~56 8. Rock in beds ... 6-00 7-87 9. Hard rock . . IO'00 13-12 The following table of velocities which 1 Hiitte ":- should not be exceeded n channels is given in the Ingenieurs Taschenbuch of the Verein Velocity. Velocity. Velocity. Slimy earth or brown clay . Clay . .... Firm sand .... Pebbly bed . . . Boulder bed . . Conglomerate of slaty fragments Stratified rocks .... . 49 98 V97 4'00 5'00 7-28 8-oo Hard rocks .. .. 14-00 36 75 1-51 3'I5 4°03 6'IO 7'45 Surface l/leimz Bottom 12- 1 5 -26 52 1-02 2-30 3-08 4°90 6-oo 10-36 l § 127. Regirne of a River Channel.-Ariver channelris said to be a state of regime, or stability, when it changes little in draught or form in a series of years. In some rivers the dee est art p p of the channel changes its position perpetually, and is seldom found in the same place in two successive years. The sinuousness of the river also changes by the erosion of the banks, so that in time the position of the river is completely altered. In other rivers the change from year to year is very small, but probably the regime is never perfectly stable except where the rivers flow over a rocky bed. If a river had a constant discharge it would gradually modify its bed till a permanent regime was established. But as the volume = < P5 °“ 1 ~= =' s"' " bu ev-:i ' " x Q ' - gl (ls. 2 3 .R.'1yn¢ 3, 596' gg =' ea ~— 7'}1n/ .-»» -n —-» 9 mf. —-» ><»~. Ai T E 3.8j'f 1f? "” """' n ltffz; ""- 'A FIG. 131. °'= I 108% 4* discharged is constantly changing, and therefore 'ff é the velocity, silt is deposited when the velocity decreases, and scour goes on when the velocity increases in the same place. When the scouring and silting are considerable, a perfect balance ~;-fe between the two is rarely established and hence e dl happen if by artificial means the erosion of the banks is prevented. If a river flows in soil incapable of resisting its tendency to scour it is necessarily sinuous (§ 107), for the slightest deflection of the current to either side begins an erosion which increases progres sively till a considerable bend is formed. If such a river ls straightened it becomes sinuous again unless its banks are protected from scour. § 128. Longitudinal Section of River Bed.-The declivity of rivers decreases from source to mouth. In their higher parts rapid and torrential, flowing over beds of gravel or boulders, they enlarge in volume by receiving affluent streams, their slope diminishes, their bed consists of smaller materials, and finally they reach the sea. Fig. IST shows the length in miles, and the surface fall in feet per mile, of the Tyne and its tributaries. The decrease of the slope is due to two causes. (1) The action of the transporting power of the water, carrying the smallest debris the greatest distance, causes the bed to be less stable near the mouth than in the higher parts of the river; and, as the river adjusts its slope to the stability of the bed by scouring or increasing its sinuousness when the slope is too great, and by silting or straightening its course if the slope is too small, the decreasing stability of the bed would coincide with a decreasing slope. (2) The increase of volume and section of the river leads to a decrease of slope; for the larger the section the less slope is necessary to ensure a given velocity. The following investigation. though it relates to a purely arbitrary case, is not without interest. Let it be assumed, to make the conditions definite-(I) that a river flows over a. bed of uniform resistance to scour, and let it be further assumed that to maintain stability the velocity of the river in these circumstances is constant from source to mouth; (2) suppose the sections of the river at all points are similar, so that, b being the breadth of the river at any point, its hydraulic mean depth is ab and its section is cb'2, where a and c are constants applicable to all parts of the river; (3) let us further assume that the discharge increases uniformly in consequence of the supply from affluents, so that, if l is the length of the river from its source to any given point, the discharge there will be A D X kl, where le is another; constant applicable to all points in the course 3 of the river. Let AB (fig. 132) be the longitudinal section of the river, whose source is at A; and take A for the origin of » vertical and horizontal coordinates. Let C be a point whose ordinates are x and y, and let the river at C have the breadth b, the slope i, and the velocity if. Since velocity >< area of section=discharge, vcb2=kl, or b =/ (kl/cv). Hydraulic mean depth =ab =a/ (kl/cv). But, by the ordinary formula for the flow of rivers, mi=§ v2; = (112/ni = (§ 'ni'/a)] (c/kl). But i is the tangent of the angle which the curve at C makes with the axis of X, and is therefore=dy/dx. Also, as the slope is small, l=AC =AD =x nearly. dy/fix = (§ 'v5/a)/ (6/kx); and, remembering that v is constant, 3/ = (25%/'¢1)~' (Cx/k): or y2=constant >< oc;, so that the curve is a common parabola, of which the axis is horizontal and the vertex at the source. This may be considered an ideal longitudinal section, to which actual rivers approximate more or less, with exceptions due to the varying hardness of their beds, and the irregular manner in which their volume increases. § 129. Surface Level of River.-The surface level of a river is a plane changing constantly in position from changes in the volume of water discharged, and more slowly from changes in the river bed, and the circumstances affecting the drainage into the river. For the purposes or the engineer, it is important to determine (1) the extreme low water level, (2) the extreme high water or flood level, and (3) the highest navigable level. I. Low I/Valet' Level cannot be absolutely known, because a river reaches its lowest level only at rare intervals, and because alterations in the cultivation of the land, the drainage, tne removal of forests, the removal or erection of obstructions in the river bed, &c., gradually alter the conditions of discharge. The lowest level of which records can be found is ta ken as the convené """Ts - Fm. 132. ""J 'Qe . 0? Q Q J

N

¢ " " continual variations occur in the form of the river "~-S and the direction of its currents. In oth r cf s e a es, where the action is less violent, a tolerable balance may be established, and the deepening of the bed by scour at one time is compensated by the silting at another. In that case the general regime is permanent, though alteration is constantly going on. This is more likely to tional or approximate low water level, and allowance is made for possible changes. g Water or Flood Level.-The engineer assumes as the highest flood level the highest level of which records can be obtained. In forming a judgment of the data available, it must be remembered that the highest level at one point of a river is not always simultaneous 2. Hiah with the attainment of the highest level at other points, and that the rise of a river in flood is very different in different parts of its course. In temperate regions, the floods of rivers seldom rise more than 20 ft. above low-water level, but in the tropics the rise of floods is greater.

3. Highest Navigable Level.-When the river rises above a certain level, navigation becomes difficult from the increase of the velocity of the current, or from submersion of the tow paths, or from the headway under bridges becoming insufficient. Ordinarily the highest navigable level may be taken to be that at which the river begins to overflow its banks.

§ 1 30. Retailer Value of Dijferent JL! aterials for Submerged Works.-That the power of water to remove and transport different materials depends on their density has an important bearing on the selection of materials for submerged works. Ill many cases, as in the aprons or flooring's beneath bridges, or in front of locks or falls, and in the formation of training walls and breakwaters by pierres perdus, which have to resist a violent current, the materials of which the structures are composed should be of such a size and weight as to be able individually to resist the scouring action of the water. The heaviest materials will therefore be the best; and the different value of materials in this respect will appear much more striking, if it is remembered that all materials lose part of their weight in water. A block whose volume is V cubic feet, and whose density in air is rp Q per cubic foot, weighs in air 'wV lb, but in water only (w-62-41) Weight of a Cub. Ft. in lb.

In Air. In Water.

Basalt . . 187-3 124-9

Brick .. .. 130-0 67-6

Brickwork .... 1 12-0 49-6

Granite and limestone 170-0 107-6

Sandstone . . 144-0 81-6

Masonry .... II6-144 | 53-6-81-6

§ 131. Inundalion Deposits from a River.-/Vhen a river carrying silt periodically overflows its banks, it deposits silt over the area flooded, and gradually raises the surface of the country. The silt is deposited in greatest abundance where the water first leaves the river. It hence results that the section of the country assumes a peculiar form, the river flowing in a trough along the crest of a ridge, from which the land slopes downwards on both sides. The silt deposited from the water forms two wedges, having their thick ends towards the river (fig. 133).

/~@” 0 Q? n%w,

Fig. 133.

This is strikingly the case with the Mississippi, and that river is now kept from flooding immense areas by artificial embankments or levees. In India, the term deltaic segment is sometimes applied to that portion of a river running through deposits formed by inundation, and having this characteristic section. The irrigation of the country in this case is very easy; a comparatively slight raising of the river surface by a weir or annicut gives a command of level which permits the water to be conveyed to any part of the district. § 132. Dallas.-The name delta was originally given to the Ashaped portion of Lower Egypt, included between seven branches of the Nile. It is now given to the whole of the alluvial tracts round river mouths formed by deposition of sediment from the river, where its velocity is checked on its entrance to the sea. The characteristic feature of these alluvial deltas is that the river traverses them, not in a single channel, but in two or many bifurcating branches. Each branch has a tract of the delta under its influence, and gradually raises the surface of that tract, and extends it seaward. As the delta extends itself seaward, the conditions of discharge through the different brancl1es change. The water finds the passage through one of the branches less obstructed than through the others; the velocity and scouring action in that branch are increased; in the others they diminish. The one channel gradually absorbs the whole of the water supply, while the other branches silt up. But as the mouth of the new main channel extends seaward the resistance increases both from the greater length of the channel and the formation of shoals at its mouth, and the river tends to form new bifurcations AC or AD (fig. 134), and one of these may in time become the main channel of the river.

§ 133. Field Operations preliminary to a Study of River Improvement.-There are required (1) a plan of the river, on which the positions of lines of levelling and cross sections are marked; (2) a longitudinal section and numerous cross sections of the river; (3) a series of gaugings of the discharge at different points and in different conditions of the river.-Longitudinal Section.-This requires to be carried out with great accuracy. A line of stakes is planted, following the sinuosities of the 1

E

river, and chained and levelled. The cross sections are referred to the line of stakes, both as to position and direction. The determination of the surface slope is very difficult, partly from its extreme smallness, partly from oscillation of the water. Cunningham recommends that the slope be taken in a length of 2000 ft. by four simultaneous observations, two on each side of the river. § 134. Cross Sections -A stake is planted flush with the water, and its level relatively to some point on the line of levels is determined. Then the depth of the water is determined at a series of points (if A ~.-.

//»

1 '°v.

D

/l

X .

B

cal/.

Sh ai;

Fig. 134.

possible at uniform distances) in a line starting from the stake and perpendicular to the thread of the stream. To obtain these, a wire may be stretched across with equal distances marked on it by hanging tags. The depth at each of these tags may be obtained by a light wooden stall, with a disk-shaped shoe 4 to 6 in. in diameter. If the depth is great, soundings may be taken by a chain and weight. To ensure the wire being perpendicular to the thread of the stream, it is desirable to stretch two other wires similarly graduated, one above and the other below, at a distance of 20 to 40 yds. A number of floats being then thrown in, it is observed whether they pass the same graduation on each wire. For large and rapid rivers the cross section is obtained by sounding in the following way. Let AC (fig. 135) be the line on which soundings are required. A base line AB is measured out at right angles to AC, and ranging staves are set up at AB and at D in line with AC. A boat is allowed to drop down stream, and, at the moment it comes in line with AD, the lead is

dropped, and an observer in the

boat takes, with a box sextant. If; the angle AEB subtended by

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ABE. By repeating observations a number of soundings are obtained, which can be plotted in their proper position, and the form of the river bed drawn by connecting the extremities of the lines. From the section can be measured the sectional area of the stream S2 and its wetted perimeter X; and from these the hydraulic mean depth rn can be calculated.

§ 135. Measurement of the Discharge of Rivers.-The area of cross section multiplied by the mean velocity gives the discharge of the stream. The height of the river with reference to some fixed mark should be noted whenever the velocity is observed, as the velocity and area of cross section are different in differ est states of the river. To determine the mean velocity various methods may be adopted; and, since no method is free from liability to error, either from the difficulty of the observations or from uncertainty as to the ratio of the mean velocity to the velocity observed, it is desirable that more than one method should be used.

TNSTRUMENTS ron l'lEASUR1NG THE VELOCITY or WATER § 136. Surface Floats are convenient for determining the surface velocities of a stream, though their use is difficult near the banks. The floats may be small balls of wood, of wax or of hollow metal, so loaded as to float nearly flush with the water surface. To render them visible they may have a vertical painted stem. In experiments on the Seine, cork balls 1% in. diameter were used, loaded to float flush with the water, and provided with a stem. In A. J. C. Cunningham's observations at Roorkee, the floats were thin circular disks of English deal, 3 in. diameter and l in. thick. For observations near the banks, floats I in. diameter and lf in. thick were used. To render them visible a tuft of cotton wool was used loosely hxed in a hole at the centre.

The velocity is obtained by allowing the float to be carried down, and noting the time of passage over a measured length of the stream. If 2' is the velocity of any float, 2 the time of passing over a length I, then v=l/t. To mark out distinctly the length of stream over which the floats pass, two ropes may be stretched across the stream at a distance apart, which varies usually from 50 to 250 ft., according to the size and rapidity of the river. In the Roorkce experiments a length of run of 50 ft. was found best for the central two-fifths of the width, and 25 ft. for the remainder, except very close to the banks, where the run was made 12% ft. only. The longer the run the less is the proportionate error of the time observations, but on the other hand the greater the deviation of the floats from a straight course parallel to the axis of the stream. To mark the precise position at which the floats cross the ropes, Cunningham used short white rope pendants, hanging so as nearly to touch the surface of the water. In this case the streams were 80 to 180 ft. in width. In wider streams the use of ropes to mark the length of run is impossible, and recourse must be had to box sextants or theodolites to mark the path of the floats. Let AB (fig. 136) be a measured base line strictly parallel to the thread of the stream, and AAI, BB1 lines at right angles to AB marked out by ranging rods at A1 and Bl. Suppose observers stationed at A A rl C, Il; A, and B with sextants or theodolites, and <f?°'°ff";"'T°"f'°' "'°°°'9 let CD be the path of any float down

§ , ' stream. As the float approaches AA1,

1 / 1 the observeratB keepsit on the cross wire

I/, of his instrument. The observer at A
~y, if observes the instant of the float reaching

| fl, J' the line AAi, and signals to B who then llil' ' reads oil the angle ABC. Similarly, as | I/l' l the float approaches BB1, the observer

, ll ' at A keeps it in sight, and when signalled

illl K to by B reads the angle BAD. The data 1 If l so obtained are sufficient for plotting 1 I/ the path of the float and determining 2 / 1 ' the distances AC, BD.

<|§ se i >, , {, The time taken by the float in passing B ll 1 .D; fill BJ over the measured distance may be ob-I, l served by a chronograph, started as the " float passes the upper rope or line, and FIG. 136. stopped when it passes the lower. In Cunningham's observations two chronometers were sometimes used, the time of passing one end of the run being noted on one, and that of passing the other end of the run being noted on the other. The chronometers were compared immediately before the observations. In other cases a single chronometer was used placed midway of the run. The moment of the floats passing the ends of the run was signalled to a timekeeper at the chronometer by shouting. It was found quite possible to count the chronometer beats to the nearest half second, and in some cases to the nearest quarter second. § 137. Sub-surface Floats.~The velocity at different depths below the surface of a stream may be obtained by sub-surface floats, used precisely in the same way as surface floats. The most usual arrangement is to have a large float. of slightly greater density than water, very light surface float. The motion connected with a small and

/ of the combined arrangement is not Egg S RT sensibly different from that of the large §§ : "i § ?';".5T'-F float, and the small surface float enables " T" “' ' ' an observer to note the path and velo-I city of the sub-surface float. The in; strument is, however, not free from | objection. If the large submerged | float is made of very nearly the same | density as water, then it is liable to be thrown upwards by very slight edclies in the water, and it does not maintain its position at the depth at which it is M ...W intended to float. On the other hand, »»lll' *f., if the large float is made sensibly i ii' llTllri, l§§ heavier than water, the indicating or § "§ ll, fM#;l»| 'l'3 surface float must be made rather large, fl” ' " ill and then it to some extent influences the motion of the submerged float.

FIG. 137. Fig. 137 shows one form of subsurface float. It consists of a couple

of tin plates bent at a right angle and soldered together at the angle. This is connected with a wooden ball at the surface by a very thin wire or cord. As the tin alone makes a heavy submerged float, it is better to attach to the tin float some pieces of wood to diminish its weight in water. Fig. 138 shows the form of submerged float used by Cunningham. It consists of a hollow metal ball connected to a slice of cork, which serves as the surface float. § 138. Twin Floats.-Suppose two equal and similar floats (fig. 139) connected by a wire. Let one float be a little lighter and the other Then the velocity of the combined

a little heavier than water.

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will be the mean of the surface velocity and the velocity at the depth at which the heavier float swims, which is determined by the length of the connecting wire. Thus if 11, is the surface velocity and 11,1 the velocity at the depth to which the lower float is sunk, the velocity of the combined floats will be U = %<U¢'l”7-'d)-Consequently,

if v is observed, and 'vs determined by an experiment with a single float

vd=2v-fu..

According to Cunningham, the twin float gives better results than the sub-surface float.

§ 139. Velocity Rods.-Another form of float is shown in fig. 140. This consists of a cylindrical rod loaded at the lower end so as to float nearly vertical in water. A wooden rod, with a metal cap at the

bottom in which shot can be placed, answers better than anything else, and sometimes the wooden rod is made in » lengths, which can be screwed together so as to suit streams of different depths. A tuft of cotton wool at the top serves to make the float more easily visible. Such a rod, so adjusted in length that it sinks nearly to the bed of the stream, gives directly the mean velocity of the whole vertical section in which it floats. '§ 140. Re'/Jy's Current Meter.-No instrument has been so much used in

directly determining the velocity of a stream at a given point as the screw current meter. Of this there are a If dozen varieties at least. As an example of the instrument in its simplest form, Revy's meter may be selected. This is an ordinary screw meter of a larger size than /jf' ”

usual, more carefully made, and with its details carefully studied (figs. I4:I, 142). FIG 140 It was designed after experience in gauging the great South American rivers. The screw, which is actuated by the water, is 6 in. in diameter, and is of the type of the Griffiths screw used in ships. The hollow spherical boss serves to make the weight of the screw sensibly equal to its displacement, so that friction is much reduced. On the axis aa of the screw is a worm which drives the counter. This consists of two worm wheels g and h fixed on a common axis. The worm wheels are carried on a frame attached to the pin l. By means of a string attached to I they can be pulled into gear with the worm, or dropped out of gear and stopped at any instant. A nut m can be screwed up, if necessary, to keep the counter permanently in gear. The worm is two-threaded, and the worm wheel g has 200 teeth. Consequently it makes one rotation for Ioo rotations of the screw, and the number of rotations up to IO0 is marked by the passage of the graduations on its edge in front of a fixed index.. The second worm wheel has 196 teeth, and its edge is divided into 49 divisions. Hence it falls behind the first wheel one division for a complete rotation of the latter. The number of hundreds of rotations of the screw are therefore shown by the number of divisions on h passed over by an index fixed to g. One difficulty in the use of the ordinary screw meter is that particles of grit, getting into the working parts, very sensibly alter the friction, and therefore the speed of the meter. Revy obviates this by enclosing the counter in a brass box with a glass face. This box is filled with pure water, which ensures a constant coefficient of friction for the rubbing parts, and prevents any mud or grit finding its way in. In order that the meter may place itself with the axis parallel to the current, it is pivoted on a vertical axis and directed by a large vane shown in fig. 142. To give the vane f »

7

f

/

f 4/

/ 7

fa/

fy/lrw 1 didn*

1/, ,,

»f

W

I

fl /

iff

f

/ more directing power the vertical axis is nearer the screw than in ordinary meters, and the vane is larger. A second horizontal vane is attached by the screws x, x, the object of which is to allow the meter to rest on the ground without the motion of the screw being interfered with. The string or wire for starting and stopping the meter is

FIG. 141. carried through the centre of the vertical axis, so that the strain on it may not tend to pull the meter oblique to the current. The pitch of the screw is about 9 in. The screws at x serve for filling the meter with water. The whole apparatus is fixed to a rod (fig. 142), of a length proportionate to the depth, or for very great depths it is fixed to a weighted bar lowered by ropes, a plan invented by Revy. The instrument is generally used thus. The reading of the counter is —g noted, and it is put out of gear. The meter is 0 then lowered into the Water to the required position from a platform between two boats, or better from a temporary bridge. Then the counter 1S put into gear for one, two or five minutes. Lastly, the instrument is raised and the counter again read. The velocity is 0 deduced from the number of rotations in unit time by the formulae given below. For surface velocities the counter may be kept permanently in gear, the screw being started " and stopped by hand. § 141. The Harlacher Current Meter.-In this the ordinary counting apparatus is abandoned. A worm drives a worm wheel, which makes an electrical contact once for each 100 rotations of the worm. This contact gives a signal above water. With this arrangement, a series of velocity observations can be made, without removing the instrument from the water, and a number of practical difficulties attending the accurate starting and stopping of the ordinary counter are entirely got rid of. Fig. 143 shows the meter. The worm wheel z makes one rotation for 100 of the screw. A pin moving the lever x makes the 1 electrical contact. The wires b, c are led through a gas pipe B; this also serves to ° ~ adjust the meter to any required position on the wooden rod dd. The rudder or vane is shown at WH. The galvanic current acts on the electromagnet rn, which is fixed in a small metal box containing also the battery. The magnet exposes and withdraws a coloured disk at an opening in the cover of the box. § 142. Arnsler Lafon Current Meter.-A very convenient and accurate current meter is constructed by Amsler Laffon of Schaffhausen. This can be used on a rod, and put into and out of gear by a ratchet. The peculiarity in this case is that there is a double ratchet, so that one pull on the string puts the counter into gear and a second puts it out of gear. The string may be slack during the action of the meter, and there is less uncertainty than when the i - ll. I FIG. 142. counter has to be held in gear. For deep streams the meter A is suspended by a wire with a heavy lenticular weight below (fig. 144). The wire is payed out from a small winch D, with an index showing the depth of the meter, and passes over a pulley B. The meter is in gimbals and is directed by a conical rudder which keeps it facing the stream with its axis horizontal. There is an electric circuit from a battery C through the meter, and a contact is made closing the circuit every 100 revolutions. The moment the circuit closes a bell rings. By a subsidiary arrangement, when the foot of the instrument, 0-3 metres below the axis of the meter, touches the ground the circuit is also closed and the bell rings. It is easy to distinguish the continuous ring when the ground is reached from the short ring when the counter signals. A convenient winch for the wire is so graduated that if ll l""'t 5* Wig., fb b c u 4§ r c l.;1..;; "V N .., ..... <4 », V st r as '/ 4,12 B, .ll

V A d V

fr/ I r»““» 'Qi “ = L/ A 1. ' Fl A f W 1 / . . H - - | »= - = I. ' , . 'f 4 FIG. 143. set when the axis of the meter is at the water surface it indicates at any moment the depth of the meter below the surface. Fig. 144 shows the meter as used on a boat. It is a very convenient instrument for obtaining the velocity at different depths and can also be used as a sounding instrument. § 143. Determination of the Coejiciients of the Current Meter.-Suppose a series of observations has been made by towing the meter in still water at different speeds, and that it is required to ascertain from these the constants of the meter. If 71 is the velocity of the watergand n the observed number of rotations per second, let '-11 = a -l-Bn (1) where a and B are constants. Now let the meter be towed over a measured distance L, and let N be the revolutions of the meter and t the time of transit. Then the speed of the meter relatively to the water is L/t=fv feet per second, and the number of revolutions per second is N/t=n. Suppose rn observations have been made in this way, furnishing corresponding values of 1: and n, the speed in each trial being as uniform as possible, 27t=7t1 - 21J='v1-l-112+ 2nv=n1°v1+n2v2+ » 2n”=ni-l-115+ - ~-En]2=[r11+n2+ - 32 In a few cases the constants for screw current meters have been


Then for the determination of the constants a and /S in (1), by the method of least squaresa

3

En2§ .'v - Eninzf

mZn2 - [21:12

mEn'v -211211

m21z2 - [En]2

stream and to check oscillations of the water column. Let the difference of level of a pair of tubes A and B (fig. 145) be taken to be h=kv2/2g, then k may be taken to be a corrective coefficient whose value in well-shaped instruments is very nearly unity. By placing his instrument in front of a boat towed through water Darcy found k=1-034; by placing the instrument in a stream the velocity of which had been ascertained by floats, he found k = I -006; by readings determined by towing them in R. E. Froude's experimental tank in taken in different parts of the section of a canal in whichaknown volume of water was flowing, h

h

C' D

B i e.. lllffiir

|~l|[f

l:f~ li: ' J fl i~ 1

/ .5 nl.. . 11"

-'-1/A Y?

J,

li W

1

/1

f ?

- xssssv

i

I

L, . v " ' "", s- 1 - ' ' // / “, ~ <1 "1 1;;§ ': $"' § / 3.4/§ {¢'Z

e found k=O-993. He believed the first value to be too high in consequence of the disturbance caused

by the boat. The mean of the other two values is almost exactly unity (Recherches hydmuliques, Darcy and Bazin, 1865, p. 63). W. B. Gregory used somewhat differently formed Pitot tubes for which the k = 1 (Am. Soc. Mech. Eng., 1903, 25). T. E. Stanton used a Pitot tube in determining the velocity of an air current, and for his instrument he found k= 1-030 to k= I~032 (“ On the Resistance of Plane Surfaces in a

Current of Air, ” Proc. Inst. Civ. Eng., 1904, 156).

One objection to the Pitot tube in its original form was the great difficulty and inconvenience of reading the height h in the immediate neighbourhood of the stream

surface. This is obviated in the Darcy gauge, which can be removed from the stream to be read.

Fig. 146 shows a Darcy gauge.

It consists of two Pitot tubes

having their mouths at right angles. In the instrument shown, the two tubes, formed of copper in the

lower part, are united into one for strength, and the mouths of the tubes open vertically and horizontally. The upper part of the tubes

is of glass, and they are provided with a brass scale and two verniers b, b. The whole instrument is supported on a vertical rod or small pile AA, the fixing at B permitting the instrument to be adjusted to any height on the rod, and at the same time allowing free rotation, so that it can be held parallel to the current. At c is a two-way cock, which can be opened or closed by cords. If this is shut, the instrument can be lifted out of the stream for reading. The glass tubes are connected at top by a brass fixing, with a stop cock a, and a flexible tube and FIG. 144.

which the resistance of ship models is ascertained. In that case the data are found with exceptional accuracy. § 144. Darcy Gauge or modzjied Pitot Tube.-A very old instrument for measuring velocities, invented by Henri Pitot in 1730 (Hisfoire de l'1-lcadémie des Sciences, 1732, p. 376), consisted simply of a vertical glass tube with a right-angled bend, placed so that its mouth was normal to the direction of flow (fig. 145). The impact of the stream on the mouth of the tube balances a column in the tube, the height of which is approximately h='U2/2g, where v is the velocity

(Vg 51| at the depth x. Placed

li.;;| | l with its mouth parallel % ~:Hi'i l;fL?w ;', 1::; ':, V A: to the stream the water

{ 21.3
§ i;{';§ !» § ii* inside the tube is nearly

“illfi '.l;'1' at the same level as the lt. 5 Il# , ' surface of the stream, V gli" i il | V and turned with the !'**> .li l ' Q mouth down stream, the |§ , 1 [il 1 fluid sinks a depth it .» 1, 1 l Q h' ='v*/2g nearly, though / Q, i' I the tube in that case "" ' '~~ Y interferes with the free A B C flow of the liquid and

FIG 145 sompwhal; modifies éhg

resu t. itot expan e

the mouth of the tube so as to form a funnel or bell mouth. In that case he found by experiment

h = 1 -522/Qg.

But there is more disturbance of the stream. Darcy preferred to make the mouth of the tube very small to avoid interference with the mouthpiece m. The use of this is as follows. If the velocity is required at a point near the surface of the stream, one at least of the water columns would be below the level at which it could be read. It would be in the copper part of the instrument. Suppose then a little air is sucked out by the tube m, and the cock a closed, the two columns will be forced up an amount correspond» ing to the difference between atmospheric pressure and that in the tubes. But the difference of level will remain unaltered. When the velocities to be measured are not very small, this instrument is an admirable one. It requires observation only of a single linear quantity, and does not require any time observation. The law connecting the velocity and the observed height is a rational one, and it is not absolutely necessary to make any experiments on the coefficient of the instrument. If we take 'u=k/ (Qgh), then it appears from Darcy's experiments that for a well-formed instrument k does not sensibly differ from unity. It gives the velocity at a definite point in the stream. The chief difficulty arises from the fact that at any given point in a stream the velocity is not absolutely constant, but varies a little from moment to moment. Darcy in some of his experiments took several readings, and deduced the velocity from the mean of the highest and lowest. § 145. Perradil Hydrodynamometer.-This consists of a frame abcd (fig. 147) placed vertically in the stream, and of a height not less than the stream's depth. The two vertical members of this frame are connected by cross bars, and united above water' by a circular bar, situated in the vertical plane and carrying a horizontal graduated circle ef. This whole system is movable round its axis, being suspended on a pivot at g connected with the fixed support mn. Other horizontal arms serve as guides. The central vertical rod gr forms a torsion rod, being fixed at r to the frame abcd, and, passing freely upwards through the guides, it carries a horizontal needle moving over the graduated circle ef. The support g, which carries the apparatus, also receives in a tubular guide the end of the torsion rod gr and a set screw for fixing the upper end of the torsion rod when necessary. The impulse of the stream of water is received on a circular disk x, in the plane of the torsion rod and the frame abcd. To raise and lower the a paratus easily, it is not fixed directly to the rod mn, but to a tube k/Qsliding on mn. Suppose the apparatus arranged so that the disk x is at that level the stream where the velocity is to be determined. The plane


abcd is placed parallel to the direction of motion of the water. Then the disk x (acting as a rudder) will place itself parallel to the stream on the down stream side of the frame. The torsion rod will be unstained, and the needle will be at zero on the graduated circle. If then the instrument is turned by pressing the needle till the lane Boileau ex eriments on v, ~ » » P

abcd of the disk and the zero of the graduated circle is at right angles to the stream, the torsion rod will be twisted through an anglle which measures the normal impulse of the stream on the disk x. T at angle forces in the torsion rod is

will be given by the distance of the needle from zero. Observation shows that the velocity of the water at a given point is not constant. It varies between limits more or less wide. /Vhen the apparatus is nearly in its right position, the set screw at g is made to clamp the torsion spring. Then the needle is fixed, and the apparatus carrying the graduated circle oscillates. It is not, then, difficult to note the mean angle marked by the needle.

Let r be the radius of the torsion, - rod, l its length from the needle I over ef to r, and a the observed

torsion angle. Then the moment

of the couple due to the molecular

"1

UI

f

£1,

M =E, Ia/l; J

where E, is the modulus of elasticity for torsion, and I the polar

moment of inertia of the section of the rod. If the rod is of circular section, I=%7|'T4. Let R be the

radius of the disk, and b its

leverage, or the distance of its

centre from the axis of the torsion rod. The moment of the pressure

of the water on the disk is

Fb = kb(G/2g)1rR21J2,

where G is the heaviness of water and k an experimental coefficient. Eila/Z = kb(G/zg)-/rR2v2.

For any given instrument, 5

v =c/ a,

where c is a constant coefficient for the instrument.

The instrument as constructed had three disksvwhich could be used at will. Their radii and leverages were in feet R = 1, =

i

l

l

° ' '.£ .z;;';;L

Then

"

W' >w F 2399 1.;§ s/$'.¥§ l“'“'5»Z%§ °> FIG. 147.

1st disk . . o-052 o-x6

2nd, , o-105 0-32

3rd, , . O'2IO o-66

For a thin circular plate, the coefficient k=I'I2J In the actual instrument the torsion rod was a brass wire o-06 in. diameter and 6% ft. long. Supposing a. measured in degrees, we get by calculation v=o~335/a; o-1154 a; o-o42/a.

Very careful experiments were made with the instrument. It was fixed to a wooden turning bridgexrevolving over a circular channel of 2 ft. width, and about Z6 ft. 'circumferential length. An allowance was made for the slight current produced in the channel. These experiments gave for the coefficient c, in the formulav =rx/ a, 1st disk, c=o~3126 for velocities of 3 to 16 ft. 2nd H 0'1177 vv H Ii to rv

3rd n 0°O349 u rv less than 1% n

The instrument is preferable to the current meter in giving the velocity in terms of a single observed quantity, the angle of torsion, while the current meter involves the observation of two quantities, the number of rotations and the time. The current meter, except in some improved forms, must be withdrawn from the water to read the result of each experiment, and the law connecting the velocity and number of rotations of a current meter is less well-determined than that connecting the pressure on a disk and the torsion of the wire of a hydro dynamo meter.

The Pitot tube, like the hydro dynamo meter, does not require a time observation. But, where the velocity is a varying one, and consequently the columns of water in the Pitot tube are oscillating, there is room for doubt as to Whether, at any given moment of closing the cock, the difference of level exactly measures the impulse of the stream at the moment. The Pitot tube also fails to give measurable indications of very low velocities. PROCESSES FOR GAUGING STREAMS

§ 146. Gauging by Observation of the Maximum Surface Velocity.-The method of gauging which involves the least trouble is to determine the surface velocity at the thread of the stream, and to degluce from it the mean velocity of the whole cross section. The maximum surface velocity may be determined by floats or by a current meter. Unfortunately the ratio of the maximum surface' to the mean velo» city is extremely variable. Thus putting U., for the surface velocity at the thread of the stream, and Um for the mean velocity of the whole cross section, 'vm/vp has been found to have the following values:- v, ,, /110

De Prony, experiments on small wooden channels 0-8164 Experiments on the Seine ....... 0-62 Destrem and De Prony, experiments on the Neva 0-78 D Canals ...... o-82

Baumgartner, experiments on the Garonne . . 0-80 Brunings (mean) ........ . 0-85

Cunningham, Solaniaqueduct. . . . . 0-823 Various formulae, either empirical or based on some theory of the vertical and horizontal velocity curves, have been proposed for determining the ratio vm, /vo. Bazin found from his experiments the empirical expression

r/, ,, =v..-25-4w/ (mi);

where m is the hydraulic mean depth and i the slope of the stream. In the case of irrigation canals and rivers, it is often important to determine the discharge either daily or at other intervals of time, while the depth and consequently the mean velocity is varying. Cunningham (Roorkee Prof. Papers, iv. 47), has shown that, for a given part of such'a stream, where the bed is regular and of permanent section, a simple formula may be found for the variation of the central surface velocity with the depth. When once the constants of this formula have been determined by measuring the central surface velocity and depth, in different conditions of the stream, the surface velocity can be obtained by simply observing the depth of the stream, and from this the mean velocity and discharge can be calculated. Let z be the depth of the stream, and to the surface velocity, both measured at the thread of the stream. Then 2/02=CZ§ where c is a constant which for the Solani aqueduct had the values 1~9 to 2, the depths being 6 to 10 ft., and the velocities 3; to 4% ft. Without any assumption of a formula, however, the surface velocities, or still better the mean velocities, for different conditions of the stream may be plotted on a diagram in which the abscissae are depths and the ordinates velocities. The continuous curve through points so found would then always give the velocity for any observed depth of the stream, without the need of making any new float or current meter observations.

§ 147. Mean Velocity determined by observing a Series of Surface Velacities.-The ratio of the mean velocity to the surface velocity in one longitudinal section is better ascertained than the ratio of the central surface velocity to the mean velocity of the whole cross section. Suppose the river divided into a number of compartments by equidistant longitudinal planes, and the surface velocity observed in each compartment. From this the mean velocity in each compartment and the discharge can be calculated. The sum of the partial discharges will be the total discharge of the stream. When wires or ropes can be stretched across the stream, the compartments can be marked out by tags attached to them. Suppose two such ropes stretched across the stream, and floats dropped in above the upper rope. By observing within which compartment the path of the float lies, and noting the time of transit between the ropes, the surface velocity in each compartment can be ascertained The mean velocity in each compartment is 0-85 to 0-91 of the surface velocity in that compartment. Putting k for this ratio, and vt, is . . . for the observed velocities, in compartments of area Sh, Q2 . then the total discharge is

Q=}<'fQ1'll1'l'92U2'l' . . .

If several floats are allowed to pass over each compartment, the mean of all those corresponding'to one compartment is to be taken as the surface velocity of that compartment. This method is very applicable in the case of large streams or rivers too wide to stretch a rope across. The paths of the floats are then ascertained in this way. Let fig. 148 represent a portion of the river, which should be straight and free from obstructions. Suppose a base line AB measured

parallel to the thread of the stream,

il 1, , l il ' and let the mean cross section of if 1 1, ' the stream be ascertained either by “L . C ..-L . sounding the terminal cross sections f 1 E b .

Q Qi, , f r 'U AE, BF, or by sounding a series of { 'Q t 'N ' equidistant cross sections. The I fl f /, 1 cross sections are taken at right , 1 1 i, 1 7 .

1 1, / V, angles to the base line. Qbservers 1 "ij, ., ' are placed at A and B with theo; 7 v, If . dolites or box Sextants. The floats

it/f I 7, are dropped in from a boat above

t 'V/| ', - 2* AE, and picked up by another boat E li, "'. " l., below BF. An observer with a /', 1 ' (lf chronograph or watch notes the /.' ' time in which each float asses P

¢ / 1 .A ' |, ill from AE to BF. The method of 7/ ill, fl Ri .7, N proceeding is this. The observer B" "" lj' 1 "" " ii"|'77tr'~"'F A sets his theodolite in the direc- fi* tion AE, and gives a signal to drop

HM a float. B keeps his instrument

r F ' on the float as it comes down. At

FK; 143 the moment the float arrives at

C in the line AE, the observer at

A calls out. B clamps his instrument and reads off the angle ABC, and the time observer begins to note the time of transit. B now points his instrument in the direction BF, and A keeps the float on the cross wire of his instrument. At the moment the float arrives at D in the line BF, the observer B calls out, A clamps his instrument and reads off the angle BAD, and the time observer notes the time of transit from C to D. Thus all the data are determined for plotting the path CD of the float and determining its velocity. By dropping in a series of floats, a number of surface velocities can be determined. When all these have been plotted, the river can be divided into convenient compartments. The observations belonging to each compartment are then averaged, and the mean velocity and discharge calculated. It is obvious that, as the surface velocity is greatly altered by wind, experiments of this kind should be made in very calm weather.

The ratio of the surface velocity to the mean velocity in the same vertical can be ascertained from the formulae for the vertical velocity curve already given (§ 101). Exner, in Erbkarrfs Zeitschrift for 1875, gave the following convenient formula. Let '11 be the mean and V the surface velocity in any given vertical longitudinal section, the depth of which is h

v/V = (1 +0-14784 h)/(1 +o~2216/ h).

If vertical velocity rods are used instead of common floats, the mean velocity is directly determined for the vertical section in which the rod floats. No formula of reduction is then necessary. The observed velocity has simply to be multiplied by the area of the compartment to which it belongs.

§ 148. Mean Velocity of the Stream from a Series of Mid Depth Velocities.-In the gaugings of the Mississippi it was found that the mid depth velocity differed by only a very small quantity from the mean velocity in the vertical section, and it was uninfluenced by wind. If therefore a series of mid depth velocities are determined by double floats or by a current meter, they may be taken to be the mean velocities of the compartments in which they occur, and no formula of reduction is necessary. If floats are used, the method is precisely the same as that described in the last paragraph for surface floats. The paths of the double floats are observed and plotted, and the mean taken of those corresponding to each of the compartments into which the river is divided. The discharge is the sum of the products of the observed mean mid depth velocities and the areas of the compartments.

§ 149. P. P. Boileau's Process for Gauging Streams.-Let U be the mean velocity at a given section of a stream, V the maximum velocity, or that of the principal filament, which is generally a little below the surface, W and w the greatest and least velocities at the surface. The distance of the principal filament from the surface is generally less than one-fourth of the depth of the stream; W is a little less than V; and U lies between W and w. As the surface velocities change continuously from the centre towards the sides there are at the surface two filaments having a velocity equal to U. The determination of the position of these filaments, which Boileau terms the gauging filaments, cannot be effected entirely by theory. But, for sections of a stream in which there are no abrupt changes of depth, their position can be very approximately assigned. Let A and l be the horizontal distances of the surface filament, having the velocity W, from the gauging filament, which has the velocity U, and from the bank on one side. Then

A/l=c“/ {(/V+2'w)/7(W-'w)},

c being a numerical constant. From gaugings by Humphreys and Abbot, Bazin and Baumgarten, the values C=O*9I9, 0-922 and O'Q25 are obtained. Boileau adopts as a mean value O'922. Hence, if /V and w are determined by float gauging or otherwise, A can be found, and then a single velocity observation at A ft. from the filament of maximum velocity gives, without need of any reduction, the mean velocity of the stream. More conveniently W, w, and U can be measured from a horizontal surface velocity curve, obtained from a series of float observations.

§ 150. Direct Determination of the lllean Velocity by a Current Meter or Darcy Gauge.-The only method of determining the mean velocity at a cross section of a stream which involves no assumption of the ratio of the mean velocity to other quantities is this-a plank bridge is fixed across the stream near its surface. From this, velocities are observed at a sufficient number of points in the cross Section of the stream, evenly distributed over its area. The mean of these is the true mean velocity of the stream. In Darcy and Bazin's experiments on small streams, the velocity was thus observed at 36 points in the cross section.

When the stream is too large to fix a bridge across it, the observations may be taken from a boat, or from a couple of boats with a gangway between them, anchored successively at a series of points across the width of the stream. The position of the boat for each series of observations is fixed by angular observations to a base line on shore.

§ ISI. A. R. Harlacher's Graphic Metliod of determining the Diss charge jrorn a Series of Current Meter Obsert/ations.—Let ABC (fig. 149) be the cross section of a river at which a complete series of A I H III IV V ' B

le /

e

1 11 } ' /

eamkw

www'/ff' W' if

”i@///.M/O / ”

FIG. 149.

current rneter observations have been taken. Let I., II., Ui . . . be the verticals at different points of which the velocities were mmsured. Suppose the depths at I., II., III., . . (fig. 149), set off as vertical ordinates in fig. 150, and on these vertical ordinates suppose the velocities set off horizontally at their proper depths. Thus, if v is the measured velocity at the de th h from the surface in fig. 149, on vertical marked III., then at lil. in fig. 150 take cd=h and ac-=v. Then d is a point in the vertical velocity curve for the vertical III., 1 and, all the velocities for that ordinate being similarly set off, the curve can be drawn. Suppose all the vertical velocity curves I .... V. (fig. ISO), thus drawn. On each of these figures draw verticals corresponding to velocities

of x, 2x, 3x ft.

per second. Then for

instance cd at III. (fig.

150) is the depth at

which a velocity of 2x

ft. per second existed

on the vertical III. in

fig. 149 and if cd is set

off at III. in fig. 149 it

gives a point in a curve

passing through points of the section where the velocity was 2x ft. per second. Set off on each of the verticals in fig. 149 all the depths thus found in the corresponding diagram in fig. 150. Curves drawn through the corresponding points on the verticals are curves of equal velocity.

The discharge of the stream per second may be regarded as a solid having the cross section of the river (fig. 149) as a base, and cross 1 II III C rv V

ll, "'f

mi) 1. Ul

FIG. ISO.;

out in this way. The upper figure shows the section of the river and the positions of the verticals at which the soundings and gaugings were taken. The lower gives the curves of equal velocity, worked out from the current meter observations, by the aid of vertical velocity curves. The vertical scale in this figure is ten times as great as in the other. The discharge calculated from the contour curves is 14-1087 cubic metres per second. In the lower figure some other interesting curves are drawn. Thus, the uppermost dotted curve is the curve through points at which the maximum velocity was found; it shows that the maximum velocity was always a little below the surface, and at a greater depth at the centre than at the sides. The next curve shows the depth at which the mean velocity for each vertical was found. The next is the curve of equal velocity corresponding to the mean Velocity of the stream; that is, it passes through points in the cross section where the velocity was identical with the mean velocity of the stream. HYDRAULIC MACHINES

§ 152. Hydraulic machines may be broadly divided into two classes: (1) M otors, in which water descending from a higher to a lower level, or from a higher to a lower pressure, gives up energy which is available for mechanical operations; (2) Pumps, in which the energy of a steam engine or other motor is expended in raising Water from a lower to a higher level. A few machines such as the ram and jet pump combine the functions of motor


Le £ bank, ,-

I Maximum surface vel, ocLt1]=V=1-352 H19f”f*'°~"k < ” ~, y ~S

E5 rvgégkxg } ? 3 r is-W .5 6? a 9 r, in '“°®1, |1i@ 'J é~ & 5 is i if -=' -+~ ' 11 <5 fp . ="» ~ '~ V . f, J

ju, 'és 'a S §§ § s § g EE $5 s R 2972.1 +03 4-so s-as -/~so 92+ no 1l'82]2~30 144114—8)16'92 11-so 195119-80 22-15 zzeq 24.90 27-so Discharge per Second, = Q= l4~-1087°“b"" <

Curves of eq'uafL velocity,

»¢N' Tra, nsform.a'£ion ratio 1031 Height scctlo=l=15 ggui $ 3 4 5 's 1, s 9 Q

-: ' s 3 ' ' S :

1" ' lf, ' 1, T Ei- gin*

1 l i A

1 4' ' .

ff sm

sbt M ' ' wp, .1 £35 f' ~ A

Wg, ii f ' f ...f1»>i'§ W@*

ng ; / . 1 4. ':

se -~ % - % /' luv'*assi

/& f ..... . rss" . tis / Nl? Qullmg f @1571 Vert: 3, = ” ', <<€>?" 5 ...tex

1 I ~ 1 .1aflx;

1° ~. ., W », § °'7' 40”

af');;&9 Qgggawf /ki

~ »', Q:

1 "~ ' /( Q

1 'z"~»-" -»"° g 8

°~». f “'f<f»@~f =~' 1 Q

9° 5 . -

yiqynq 03

4

FIG. 151.

sections normal to the plane of fig. 149 given by the diagrams in fig. 150. The curves of equal velocity may therefore be considered as contour lines of the solid whose volume is the discharge of the stream per second. Let S20 be the area of the cross section of the river, $21, S22 . . the areas contained by the successive curves of equal velocity, or, if these cut the surface of the stream, by the curves and that surface. Let x be the difference of velocity for which the successive curves are drawn, assumed above for simplicity at 1 ft. per second. Then the volume of the successive layers of the solid body whose volume represents the discharge, limited by successive planes passing through the contour curves, will bc %x(S2o+91). %x(S21+9¢). and S0 On-Consequently the discharge is

Q =xli(9o+9») +91 =0z+ ~~- -l"9»-1}-The areas Qo, S21 . . are easily ascertained by means of the polar planimeter. A slight difficulty arises in the part of the solid lying above the last contour curve. This will have generally a height which is not exactly x, and a form more rounded than the other layers and less like a conical frustum. The volume of this may be gzsgimaaed ieréaaatily, and tkakera to be the area of its base (the area j mu tip ie y 3 to 5 its eig t. 1

Fig. 151 shows the results of one of Harlacher's gaugings worked and pump. It may be noted that constructively pumps are essentially reversed motors. The reciprocating pump is a reversed pressure engine, and the centrifugal pump a reversed turbine. Hydraulic machine tools are in principle motors combined with tools, and they now form an important special class. Water under pressure conveyed in pipes is a convenient and economical means of transmitting energy and distributing it to many scattered working points. Hence large and important hydraulic systems are adopted in which ata central station water is pumped at high pressure into distributing mains, which convey it to various points where it actuates hydraulic motors operating cranes, lifts, dock gates, and in some cases riveting and shearing machines. In this case the head driving the hydraulic machinery is artificially created, and it is the convenience of distributing power in an easily applied form to distant points which makes the system advantageous. As there is some unavoidable loss in creating an artificial head this system is most suitable for driving machines which work intermittently (see Power; TRANSMISSION). The development of electrical methods of transmitting and distributing energy has led to the utilization of many natural waterfalls so situated as to be useless without such a means of transferring the power to points where it can be conveniently applied. In some cases, as at Niagara, the hydraulic power can only be economically developed in very large units, and it can be most conveniently subdivided and distributed by transformation into electrical energy. Partly from the development of new industries such as paper-making from wood pulp and electro-metallurgical processes, which require large amounts of cheap power, partly from the facility with which energy can now be transmitted to great distances electrically, there has been a great increase in the utilization of water-power in countries having natural waterfalls. According to the twelfth census of the United States the total amount of water-power reported as used in manufacturing establishments in that country was I,130,43I h.p. in 1870; 1,263,343 h.p. in 1890; and 1,727,258 h.p. in 1900. The increase was 8-4% in the decade 1870-1880, 31% in 1880-1890, and no less than -7% in 1890-1900. The increase is the more striking because this census the large amounts of hydraulic power which are transmitted electrically are not included. XII. IMPACT AND REACTION OF WATER § 153. When a stream of fluid in steady motion impinges on a solid surface, it presses on the surface with a force equal and opposite to that by which the velocity and direction of motion of the fluid are changed. Generally, in problems on the impact of fluids, it is necessary to neglect the effect of friction between the fluid and the surface on which it moves.

During Impact the Velocity of the Fluid relatively to the Surface on which it impinge; remains unchanged in M agnitude.-Consider a mass of fluid flowing in contact with a solid surface also in motion, the motion of both fluid and solid being estimated relatively to the earth. Then the motion of the fluid may be resolved into two parts, one a motion equal to that of the solid, and in the same direction, the other a motion relatively to the solid. The motion which the fluid has in common with the solid cannot at all be influenced by the contact. The relative component of the motion of the fluid can only be altered in direction, but not in magnitude. The fluid moving in contact with the surface can only have a relative motion parallel to the surface, while the pressure between the fluid and solid, if friction is neglected, is normal to the surface. The pressure therefore can only deviate the fluid, without altering the magnitude of the relative velocity. The unchanged common component and, combined with it, the deviated relative component give the resultant final velocity, which may differ greatly in magnitude and direction from the initial velocity.

From the principle of momentum, the impulse of any mass of fluid reaching the surface in any given time is equal to the change of momentum estimated in the same direction. The pressure between the fluid and surface, in any direction, is equal to the change of momentum in that direction of so much fluid as reaches the surface in one second. If P., is the pressure in any direction, m the mass of fluid impinging per second, -va the change of velocity in the direction of P" due to impact, then

36

in

Pa =mva.

If vi (fig. 152) is the velocity and direction of motion before impact, vg that after impact, then U is the total change of motion due to impact. The resultant pressure of the I/ fluid on the surface is i n the direction of V If in and is equal to v multiplied by the mass 1 / lI'l'lp1l'lgll'lg per second. That Is, putting f V ' P for the resultant pressure, f P = mv.

/' Let P be resolved into two components, V / N and T, normal and tangential to the 2 direction of motion of the solid on which FIG- 152- the fluid impinges. Then N is a lateral force producing a pressure on the supports of the solid, T is an eflort which does work on the solid. If u is the velocity of the solid, Tu is the work done per second by the fluid in moving the solid surface.

Let Q be the volume, and GQ the weight of the fluid impinging per second, and let vi be the initial velocity of the fluid before striking the surface. Then GQUIZ/2g is the original kinetic energy of Q cub. ft. of fluid, and the efficiency of the stream considered as an arrangement for moving the solid surface is, -rp=Tu/(GQ1112/zg).

§ 154. Jet deviated entirely in one Direction.-Geometrical Solution (fig. 153).-Suppose a jet of water impinges on a surface ac with a ve oclty ab, and let it be wholly deviated in planes parallel to the figure. Also let ae be the velocity and direction of motion of the surface. loin eb; then the water moves with respect to the surface in the direction and with the velocity eb. As this relative velocity is unaltered by contact with the surface, take ed =eb, tangent to the surface at c, then cd is the relative motion of the water with respect to the surface at c. Take of equal and parallel to ae. Then fc (obtained by compounding the relative motion of water to surface and common velocity of water and surface) is the absolute velocity and direction it

f

fa, ~.

9

Z,









g

~

"i

1

I

/ ~


1

r 1

1

('




ll

gs.", 1

iQ ';~" '//



é' 
'    ' "" Ta  ent
  1  '
 I   

gg, (1, V '

» >»»; >, ;i'>

%{4, , / /{/ /, ,, —" b


/ -5"

//f

FIG. 153.

of the water leaving the surface. Take ag equal and parallel to fc. Then, since ab is the initial and ag the final velocity and direction of motion, gb is the total change of motion of the water. The resultant pressure on the plane is in the direction gb. join eg. In the triangle gae, ae is equal and parallel to df, and ag to fc. Hence eg is equal and parallel to cd. But cd=eb=relative motion of water and surface. Hence the change of motion of the water is represented in magnitude and direction by the third side of an isosceles triangle, of which the other sides are equal to the relative velocity of the water and surface, and parallel to the initial and final directions of relative motion. SPECIAL CAsIzs

§ 155. (1) A Jet impinges on a plane surface at rest, in a direction normal to the plane (fig. 154).-Let a jet whose section is w impinge with a velocity 'v on a plane surface at rest, I

Ji

in a direction normal to the plane. The ll particles approach the plane, are gradually ill deviated, and finally flow away parallel to fr the plane, having then no velocity in the original direction of the jet. The quntity li of water impinging per second is mv. The pressure on the plane, which is equal to V I, the change of momentum per second, is !§ *'??f°;'* *T P=<G/wr.

(2) I f the, plane is moving in the direction '”', of the jet wit the velocity =I=u, the quantity ll impinging per second is w(v=+= u). The ll; momentum of this quantity before impact, is (G/g)w('v=1=u)11. After impact, the water 'l still possesses the velocity =l=u in the, direction of the jet; and the momentum, 'L in that direction, of so much water as lil im inges in one second, after impact, is

  1. FG/g)w(1»=Fu)u. The pressure on the

plane, which is the change of momentum IFIG. 154. 'V

per second, is the difference of these quantities or P = (G/g)w(v=1= u)2. This differs from the expression obtained in the previous case, in that the relative velocity of the water and plane 'vmu is substituted for 'v. The expression maybe written P =2 XG ><w(i1= 102/2g, where the last two terms are the volume of a prism of water whose section is the area of the jet and whose length is the head due to the relative velocity. The pressure on the plane is twice the weight of that prism of water. The work done when the plane is moving in the same direction as the jet is Pu: (G/g)<-1('v-u)'u foot-pounds per second. There issue from the jet wv cub. ft. per second, and the energy of this quantity before impact is (G/2g)wU3. The efhciency of the jet is therefore n=2(v-u)'u/93. The value of u which makes thisa maximum is found by differentiating and equating the differential coefficient to zero:- dv;/du=2(v2-4vu+3§ ¢**)/'v'*=o; u =v or ga. The former gives a minimum, the latter a maximum efficiency. Putting u = 51' in the expression above, 11 max. = 58, (3) I f, instead of one plane moving before the jet, a series of planes are introduced at short intervals at the same point, the quantity of water impinging on the series will be mv instead of w(°v-u), and the whole pressure=(G/g)a>v(v-u). The work done is (G/g)wvu(v-u). The efficiency 17 = (G/g)w°vu(v-u)+(G/2g)w11° =2u(v-u)/vz. This becomes a maximum for dn/du=2(v-2u) =0, or u=&v, and the 71=%. This result is often used as an approximate expression for the velocity of greatest efficiency when a jet of water strikes the floats of a water wheel. The work wasted in this case is half the whole energy of the jet when the floats run at the best speed. § 156. (4) Case nfa Jet impinging on a Concave Cug Vane. velocity of water v, velocity of vane in the same direction u (g. 155), weight impinging per second = Gw(v A u). If the cup is hemispherical, the water leaves the cup in a direction parallel to the jet. Its relative velocity is 'U-u when approaching the cup, and -(v-u) when leaving it.

Hence its absolute ve ocity

5* when leaving the cup is M, V, E u-(11-u)=2u-v. The { ; i ~:g v Www change of momentum er

  • '"""' “ ” ' second = (G/g)w(v-u) fi:-

M/ (2u~'v)} = 2(G/g)w(v-u)'. AQ(Comparing this with case 2, //' it is seen that the pressure on a hemispherical cu is 2u-v double that on a flat plane. The work done on the cup=2(G/g)w (1/-'u)'u foot pounds per second. The efficiency of the jet is greatest when v=3u; in that case the efficiency = If a series of cup vanes are introduced in front of the jet, so that the quantity of water acted upon is tw instead of a»(v-u), then the whole pressure on the chain of cups is ((i/g)w'U{U-(Zu-'U)}='2(G/g)w1/(U-M). In this case the efficiency is greatest when v=2u, and the maximum efficiency is unity, or all the energy of the water is expended on the cu s. lg 157. (5) Case of a Flat Vane oblique to the Jet (fig.156).-This case presents some difficulty. The water spreading on the plane in all Fig. 155. / D

/ Vr .' "'. )7'?)»*-> /f " '- i?;f:~~-»< -L —; ' ~' V, " B is gs.; ¢ A . .

W ¢ gp—" .f 'Ji

// /7 -"" C FIG. 156. directions from the point of impact, different particles leave the plane with different absolute velocities. Let AB=v=velocity of water, AC=u=velocity of plane. Then, completing the parallelogram, AD represents in magnitude and direction the relative velocity of water and plane. Draw AE normal to the plane and DE parallel to the plane. Then the relative velocity AD may be regarded as consisting of two components, one AE normal, the other DE parallel to the plane. On the assumption that friction is insensible, DE is unaffected by impact, but AE is destroyed. Hence AE represents the entire change of velocity due to impact and the direction of that change. The pressure on the plane is in the direction AE, and its amount is = mass of water impinging per second X AE. Let DAE =0, and let AD =-v, . Then AE='v, cos 0; DE=v, - sin 0. If Q is the volume of water impinging on the plane per second, the change of momentum is (G/g)Qv, cos 0. Let AC=u=velocity of the plane, and let AC make the angle CAE=5 with the normal to the plane. The velocity of the plane in the direction AE= u cos 5. The work of the jet on the plane= (G/g)Q'v, cos 0 u cos 5. The same problem may be thus treated algebraically (fig. 157). Let BAF =o., and CAF =5. The velocity 11 of the water may be decomposed into AF=v cos a. normal to the plane, and FB ='v sin u. parallel to the plane. Similarly the velocity of the plane =u =AC = BD can be decomposed into BG =FE =u cos 5 normal to the plane, and DG ==1¢ sin 5 parallel to the plane. As friction is neglected, the velocity of the water parallel to the plane is unaffected by the impact, but its component v cos a normal to the plane becomes after impact the same as that of change of velocity during change of momentum per 87 the plane, that is, u cos 5. Hence the impact=AE='v cos a.-u cos 5. The second, and consequently the normal (D fi " Y/ / (y " / Jil fl pressure on the plane is N = in the direction in which the FIG. 157. (G/g) Q (v cos a-ucos 5). The pressure plane is moving is P =N cos 5 = (G/g)Q (v cos o.-u cos 5) cos 5, and the work done on the plane is Pu(G/g)Q(v cos a.-u cos 5) u before, since AE ='v, - cos 9 cos 5, which is the same expression as ='v cos a-u cos 5. In one second the plane moves so that the point A (fig. 158) comes to C, or from the position shown in full lines to the Position shown in dotted ines. If the plane remained stationary, a length AB =1/ of the jet would impinge on, ,H the plane, but, since the plane A “ I/2/' B moves in the same direction /jf as the °et, only the length 5 HB =AE-AH impinged on U the plane. But AH =AC cos 5/ cos o. = u cos 5/ cos a, and therefore N HB='u-u cos 5/ cos a. Let wTsectional area of ljet; . vo ume impinging on ane per second=Q=w(v-up cos FIG' 158 5/cos a) =w(v cos a-u cos 5)/ cos a. Inserting this in the formulae above, we get N=Q l(v cos a-u cos 5)2- (I) g cos a P=Q (v cos a-ucos 5)2; (2) g cos a Pu=§ -wugikvcos G.-1uC0S5)2. (3) COS U. 3 . . .. . Three cases may be distinguished:- (a) The plane is at rest. work done on the plane and Then u=o, N = (G/g)w1/2 cos a; and the the efficiency of the -jet are zero. (b) The lane moves arallel to the jet. Then 5=a., and Pu-P P (G/g) wu cos 2a('v - u)2, which When u=§ v then Pu max.= =11=2, ' cos 2a. is a maximum when u=§ ~v. 547 (G/g)<»113 cos za, and the efficiency (c) The plane moves perpendicularly to the jet. Then 5=9o°-a; G cos 5=sin a; and Pu=-wu Z mum when u=§ v cos a. S111 U. COS 0. (v cos a-u sin a)2. This is a maxi-When u=§ v cos a, the maximum work and the efficiency are the same as in the last case. § 158. Best Form of Vane normally or obliquely on a after impact, and the work c ally lost, from the impossibili to receive Water.-When water impinges plane, it is scattered in all directions arried away by the water is then generty of dealing afterwards with streams of water deviated in so many directions. By suitably forming the vane, B » »' . "~ li, “" 5 /f ... ... -. - ..- Cz( T }T'-, f/V ss / f ' D

however, the water may be FIG. 159. entirely deviated in one direction, ana the loss of energy from agitation of the water is entirely avoided. Let AB (fig. 159) be a vane, on which a jet of water impinges at the point A and in the direction AC. Take AC=v=velocity of water, and let AD represent in magnitude and direction the velocity of the vane. Completing the parallelogram, DC or AE represents the direction in which the Water is moving relatively to the vane. If the lip of the vane at A is tangential to AE, the water will not have its direction suddenly changed when it impinges on the vane, and will therefore have no tendency to spread laterally. On the contrary it will be so gradually deviated that it will glide up the vane in the direction AB. This is sometimes expressed by saying that the vane receives the water without shock. § 159. Floats of Poncelet Water Wheels.-Let AC (fig. I6O) represent the direction of a thin horizontal stream of water having the


FIG. 160.

velocity v. Let AB be a curved float moving horizontally with velocity u. The relative motion of water and float is then initially horizontal, and equal to 'U-u.

In order that the float may receive the water without shock, it is necessary and sufficient that the lip of the Boat at A should be tangential to the direction AC of relative motion. At the end of ('v-u)/g seconds the float moving with the velocity u comes to the position AIB1, and during this time a particle of water received at A and gliding up the fioat with the relative velocity v-u, attains a height DE= (v-u)2/2g. At E the water comes to relative rest. It then descends along the float, and when after 2(v-u)/g seconds the fioat has come to A2B2 the water will again have reached the lip at A2 and will quit it tangentially, that is, in the direction CA2, with a relative velocity-(v-u) = -/ (2gDE) acquired under the influence of gravity. The absolute velocity of the water leaving the float is therefore u-(v-u) =2u-v. If u=é v, the water will drop off the bucket deprived of all energy of motion. The whole of the work of the jet must therefore have been expended in driving the float. The water will have been received without shock and discharged without velocity. This is the principle of the Poncelet wheel, but in that case the floats move over an arc of a large circle; the stream of water has considerable thickness (about 8 in.); in order to get the water into and out of the wheel, it is then necessary that the lip of the fioat should make a small angle (about 15°) with the direction of its motion. The water quits the wheel with a little of its energy of motion remaining.

§ I6O. Pressure on a Curved Surface when the Water is deviated wholly in one Direction.-When a jet of water impinges on a curved surface in such a direction that it is received without shock, the pressure on the surface is due to its gradual deviation from its first direction. On any portion of the area the pressure is equal and opposite to the force required to cause the deviation of so much water as rests on that surface. In common language, it is equal to the centrifugal force of that quantity of water. Case I. Surface Cylindrical and Stationary.~Let AB (fig. 161) be the surface, having its axis at O and its radius =r. Let the water impinge at A tangentially, 0 and quit the surface tangentially § '~' at B. Since the surface is at rest,

¢.* V v is both the absolute velocity of

in—', ' the water and the velocity relatively f B to the surface, and this remains un:

'~ changed during contact with the

1 ' surface, because the deviating force I is at each point perpendicular to the direction of motion. The water V I, C is deviated through an angle UW-'>' - —— - BCD=AOB=¢. Each particle of D water of weight p exerts radially R a centrifugal force pvg/rg. Let the FIG 161 thickness of the stream=t ft. Then the weight of water resting on

unit of surface=Gt lb; and the normal pressure per unit of surface =n=Gtv2/gr. The resultant of the radial pressures uniformly distributed from A to B will be a force acting in the direction OC bisecting AOB, and its magnitude will equal that of a force of intensity = n, acting on the projection of AB on a plane perpendicular to the direction OC. The length of the chord AB = 2r sin sip; let b=breadth of the surface perpendicular to the plane of the figure. The resultant pressure on surface 1 2 A

=R =2rb sing >< (-I-l=2Qbtv2s1n?- 2 g r g 2

which is independent of the radius of curvature. It may be inferred that the resultant pressure 15 the same for any curved surface of the sange projected area, which deviates the water through the same ang e.

Case 2. Cylindrical Surface moving in the Direction AC with Velocity u.—The relative velocity =v -u. The final velocity BF (fig. 162} is found by combining the relative velocity BD =v—u tangential to the surface with the velocity BE =u of the surface. The intensity of normal pressure, as in the last case, is (G/g)t(v—'u)”/r. The resultant 1,

N

0¢s»»£. -* 2T. ..... F

lv. - A "

1 ~~ ':,

5 . ' i (,

2 ' "4 ' .

1 ':- ' If

s L /' ¢ rr.;

AB:>>—»z

E: K “

jj ' 2, /,

A:

&FIG. 162.

normal pressure R = 2(G/g) bt(v -u)” sin may be resolved into two components other perpendicular to the direction former is an effort doing work on the force which does no work.

é ¢. This resultant pressure

P and L, one parallel and tne

of the vane's motion. The

vane. The latter is a lateral

P = R sin § 4>= (G/g)bt(v-u)2(1 -cos 4>); L = R cos § 4> = (G/g)bt(v -u)2 sin 4>. The work done by the jet on the vane is Pu= (G/g)btu(v-u)2(1cos ¢>), which is a maximum when u=§ v. This result can also be obtained by considering that the work done on the plane must be equal to the energy lost by the water, when friction is neglected. If ¢>=180°, cos ¢=-1, I'-COS ¢>=2; then P=2(G/g)bt(v-u)*, the same result as for a concave cup. § 161. Position which a Movable Plane takes in Flowing Water.-When a rectangular plane, movable about an axis parallel to one of its sides, is placed in an indefinite current of fluid, it

takes a position such that the , y~ resultant of the normal pres- ¢{ sures on the two sides of the —~¢'— — >;—/L——4—431| axis passes through the axis. g '# If, therefore, planes ivoted if!, f " so that the ratio a/b (ting. 163) . is varied are placed in water, A 'b and the angle they make with // the direction of the stream is observed, the position of the '{ resultalxt of the pressures on FIG. 163. the plane is determined for

different angular positions. Experiments of this kind have been made by Hagen. Some of his results are given in the following table .

r

I

if Larger plane. Smaller Plane. a/b=1-0 4>= ..; 4>=90°

0'9 75 72%°

0-8 ° 57:

o-

o~6

o-

o-4

0.

60°

8

35°

13°

80

3 6%°

0-2 °

43

292

I3 o

6%

4

§ 162. Direct Action distinguished from Reaction (Rankine, Steam Engine, § 147).

The pressure which a jet exerts on into two parts, viz.:-

a vane can be distinguished

(I) The pressure arising from changing the direct component of the velocity of the water into the velocity of the vane. In fig. 153, § 154, ab cos bae 1s the direct component of the water's velocity, or component in the direction of motion of vane. This is changed intcfqthe velocity ae of the vane. The pressure due to direct impulse is t en

P1 =GQ(ab cos bae-ae)/g. i »

For a flat vane moving normally, this direct action is the only action producing pressure on the vane. (2) The term reaction is applied to the additional action due to the direction and velocity w1th which the water glances off the vane. It is this which is diminished by the friction between the water and the vane. In Case 2, § 160, the direct pressure is P1 = Gbt(v -u)2/g.

That due to reaction is

P2 = -Gbt(v-u)2cos qS/g.

If ¢<90°, the direct component of the water's motion is not wholly converted into the velocity of the vane, and the whole pressure due to direct impulse is not obtained. If ¢>900, cos t, '> is negative and an additional pressure due to reaction is obtained. § 163. Jet Propeller.-In the case of vessels propelled by a jet of water (fig. 164), driven steriiwards from orifices at the side of the vessel, the water, originally at rest outside the vessel, is drawn into the ship and caused to move with the forward velocity V of the ship. Afterwards it is projected stern wards from the jets with a velocity v relatively to the ship, or

v-V relatively to the earth. If S2 is Fig. 164. A

the total sectional area of the jets, S211 is the quantity of water discharged per second. The momentum generated per second in a stern ward direction is (G/g)§ Zt'(v-V), and this is equal to the forward acting reaction P which propels the ship.

The energy carried away by the water =%(G/g)§ w(v-V)'- (I)

The useful work done on the ship PV = (C/g)S2v(v-V)V. (2)

Adding (1) and (2), we get the whole work expended on the water, neglecting friction:-

W= i(G/§)9v(v'-V')-Hence

the efficiency of the jet propeller is PV/W=2V/(if-i-V). (3)

This increases towards unity as v approaches V. In other words, the less the velocity of the jets exceeds that of the ship, and therefore the greater the area of the orifice of discharge, the greater is the efiicienc ' of the ro eller

3 P P -

In the “ Waterwitch " 11 was about twice V. Hence in this case the theoretical efficiency of the propeller, friction neglected, was about 3.

§ 161. Pressure of a Steady Stream in a Uniform Pipe an a Plane normal to the Direction of Motionf-Let CD (fig. 165) be a plane alia EAI § A=

c .f '5;1§

E—/ ' CE

/

"'T""'l' ° X Di /Q/ + i .f. —5-i-'§ %f/f//e”-fi"-'*

—i-~~&.= ¢f~-~-'-§ —lilo

iA1 U12

FIG. 165.

placed normally to the stream which, for simplicity, may be supposed to flow horizontally. The fluid filaments are deviated in front of the plane, form a contraction at A1A, , and converge again, leaving a mass of eddying water behind the plane. Suppose the section .~: taken at a point where the parallel motion has not begun to be disturbed, and AQA2 where the parallel motion is reestablished. Then since the same quantity of water with the same velocity passes AQAO, AQA2 in any given time, the external forces produce no change of momentum on the mass AQAQAQAQ, and must therefore be in equilibrium. If S2 is the section of the stream at AOAQ or ALM, and on the area of the plate CD, the area of the contracted section of the stream at A, A, will be c¢(§ Z-w), where cc is the coefficient of contraction. Hence, if 11 is the velocity at AOAQ or AQAQ, and U, the velocity at ArA, ,

vQ=c, 'u-(SZ-w);-

iz, = 1152/ca (Q - ei) . (1)

Let pn. Pi. 152 be the pressures at the three sections. Applying Bernoulli's theorem to the sections AoA0 and A1A1, ji* *KJQ if

G+2g ' G +2g.

Also, for the sections A, A, and A¢A2, allowing that the head due to the relative velocity v1~v is lost in shock 1fc alia 'if &£f.

tL+2g"L;+2g+ 2g

~'~P0-P2=G(1'1"7f)2/285 (2)

or, introducing the value in (1), G o rv,

P" P2 s 6—.<r. ..> I <2>>

Now the external forces in the direction of motion acting on the mass ADAOAQAQ are the pressures p, ,S2, -p29 at the ends, and the reaction -R of the plane on the water, which is equal and opposite to the pressure of the water on the plane. As these are in equilibrium, (Po-P2)9'R=05

- R=G$l l.—r ”””"

c¢(S'z-w) 2g

an expression like that for the pressure of an isolated jet on an indefinitely extended plane, with the addition of the term in brackets, which depends only on the areas of the stream and the plane. For a given plane, the expression in brackets diminishes as Q increases. If S2/w=p, the equation (4) beeonies <sf=»>“i

N23 p vita-1) I (40)

which is of the form

R =Gw<U2/2g)K,

vqhere K depends only on the ratio of the sections of the stream and p ane.

For example, let c, =o~85, a value which is probable, if we allow that the sides of the pipe act as internal borders to an orifice. Then K=p 1-1761-1 2.

= K:

1 oo

2 3-66

3 I'75

4 P29

5 I°IO

I0 -94

50 2~oo

ioo 3~5o

The assumption that the coefficient of contraction cr, is constant for 'different values of p is probably only true when p is not very large. Further, the increase of K for large values of p is contrary to experience, and hence it may be inferred that the assumption that all the filaments have a common velocity U1 at the section A1A1 and a common velocity 1; at the section AQA2 is not true when the stream is very much larger than the plane. Hence, in the expression R = KGwU2/2g,

K must be determined by experiment in each special case. For a cylindrical body putting w for the section, c¢ for the coefficient of contraction, c, ,(§ 2-w) for the area of the stream at AiAi, 11, =vS2/c¢(§ Z-w); iJ2='uSZ/(52-tu); or, putting p=Q/w,

vi=vn/Mn-I), v==vp/(f>-I)Then

R = K1GwU2/2g,

where 2 2 2

- P L L

K1 pi<P'1><6c I) + <P'I Ii

Taking e, =o-85 and p=4, K1=0~467, a value less than before. Hence there is less pressure on the cylinder than on the thin plane. § 165. Distribution of Pressure on a Surface on which a Jet impinges normally.-The principle of momentum gives readily enough the total or resultant pressure of a jet impinging on a plane surface, but in some cases it is useful to know the distribution of the pressure. The problem in the case in which the plane is struck normally, and the jet spreads in all directions, is one of great complexity, but even in that case the maximum intensity of the pressure is easily assigned. Each layer of water Flowing from an orifice is gradually deviated (fig. 166) by contact with the surface, and during deviation exercises

a centrifugal pressure towards the axis of the jet. The force exerted by each small mass of water is

normal to its path and inversely as FIG 166 the radius of curvature of the path. Hence the greatest pressure on the plane must be at the axis of the jet, and the pressure must decrease from the axis outwards, in some such way as is shown by the curve of pressure in fig. 167, the branches of the curve being probably asymptotic to the plane.

For simplicity suppose the jet is a vertical one. Let hr (fig. 167) be the depth of the orifice from the free surface, and -U1 the velocity of discharge. Then, if w is the area of the orifice, the quantity of water impinging on the plane is obviously Q=°~'71i=w/(2§ hi)§

that is, supposing the orifice rounded, and neglecting the coefficient of discharge.

The velocity with which the fluid reaches the plane is, however, greater than this, and may reach the value v=~/(2271);

where h is the depth of the plane below the free surface. The external layers of fluid subjected throughout, after leaving the orifice, to the atmospheric pressure will attain the velocity 11, and will flow away with this velocity unchanged except by friction. The layers towards the interior of the jet, being subjected to a pressure greater than atmospheric pressure, will attain a less velocity, and so much less as they are nearer the centre of the jet. But the pressure Fig. 168 shows the pressure curves obtained in three experiments with three jets of the sizes shown, and with the free surface level in the reservoir at the heights marked. on a brass plate 9 in. in diameter. small hole in the brass plate

of the jet on the plane is h ft. of water. If the pressure curve is l H '75 3 '73 2l'8 drawn with pressures represented by feet of water, it will touch the H '3 2'2 '73 21 free water surface at the centre of the jet. 42'15 '85 V6 1 '33 20'3 Suppose the pressure curve rotated so as to form a solid of revolu- H '95 I '83 I9'3 tion. The weight of water contained in that solid IS the total H '93 I3 pressure of the jet on the surface, which has already been deter- H ~93 17 1 mined. Let V =volume of this solid, then GV is its weight in pounds. 26'5 V13 I3'5 Consequently 77 V13 12'5 G¥=“'£'2zG”'3" ~ is “ai =2L»J 11 - P ° We have already, therefore, two conditions to be satisfied by the ' V33 8 pressure curve ' V38 7 1 Some very interesting experiments on the distribution of pressure I 8 on a surface struck by a jet have been made by ]. Beresford ' 1 4 5, (Prof. Papers on Indzan Engrneerzng, No. cccxxu.), with aview to ' I 5§ 4 3 afford information as to the forces acting on the aprons of welrs. ' I 5 3 5 Cylindrical jets 5 in. to 2 in. diameter, issuing from a vessel in "V 9 2 which the water level was constant, were allowed to fall vertically As the general form of the pressure curve has been already indicommumcated by a flexible tube with a vertical pressure column. Arrangements were made by which this aperture could be moved 215 in. at a time across the area struck by the jet. The height of the pressure column for each osition of the a erture ave th r p p, g e p essure, at that point of the area struck by the jet. When the aperture was l E -Diana.-4Bz> ' j;

i: 1 1 - /£ | I l”""'°"'2£'fff=°f=f .“"?""“'f" 9? = T L»~='f~~»§ @w°ff// “ . 1 "' I; —, ,, ,

-.— -.—..;L .—' ' . ..}.....-....-1—..T;i
°.jix 40E;
a; - . g

u f . I j I s 1 - = jf, -, I rl -ofm.1~95§ — -§ - — ~-> Q Z5; 'S - —....-, ;. | -.-. .. . ...-. .., .... -'s°; ~ 3 I j j, w-mzm/r nfséwsffju 1, E —, ,

. .
v ' “ o

If 0" ll — -i— -1 §

» V - I 1:S: 2

I | 0 —';.- -.-.< .:-&.—...é1-.-, ..- . 1 -. —.E?».... I je.. 2° 5: f | mg » ' 1 S1 is =2: »=: Iul gr; L” ' -

*i ' ' I

| K. I — ...-.-. -T .-..., —..1...—.... .... ....... .. .- -.-.-, .— ..-, ,° I

I 1 i I Q 2 1 I | |; 1 1 } 1-o ofs o o~5 1'° 1'5 ° Distance from axis of jet in inches. FIG. 168.-Curves of Pressure of jets impinging normally on a Plane. exactly in the axis of the jet, the pressure column was very nearly level with the free surface in the reservoir supplying the jet; that is, the pressure was very nearly v2/2 g. As the aperture moved away from the axis of the jet, the pressure diminished, and it became insensibly small at a distance from the axis of the jet about equal to the dia meter of the jet. Hence roughly the pressure due to the jet extends cated, it may be assumed that its equation is of the form y=ab"2- (I) But it has already been shown that for x-=o, y=h, hence a=h. To determine the remaining constant, the other condition ma b Y e used, that the solid formed by rotating the pressure curve represents the total pressure on the plane The volume of the solid is ® V= Ozrrxydx °° 2 =21rh ob zxdx 2 Q = oh/10g.1>> [-ff' jo -=-irh/log, b. Using the condition already stated, 2w/(hhi)=1rh/log Bb, log e b = (1r/2w)x/ (h/hr). Putting the value of b in (2) in eq. (1), and also r for the radius of the jet at the orifice, so that w=1rr2, the equation to the pressure curve is T22 y==he§ li;- § 166. Resistance of a Plane moving through a Fluid, or Pressure of a Current on 0, Plane.-When a thin plate moves through the air, or through an indefinitely large mass of still water, in a direction normal to its surface, there is an excess of pressure on the anterior face and a diminution of pressure on the posterior face. Let vbe the relative velocity of the plate and fluid, SZ the area of the plate, G the density of the fluid, h the height due to the' velocity, then the total resistance is expressed by the equation R =fG§ 2 D2/2g pounds ==fG§ 2h; where f is a coefficient having about the value 1 -3 for a plate moving in still fluid, and 1»8 for a current impinging on a fixed plane, whether the fluid is air or water. The difference in the value of the coefficient in the two cases is perhaps due to errors of experiment. There is a similar resistance to motion in the case of all bodies of “ unfair ” form, fthat is, lin which the surfaces over which the water slides are t d | no o gra ua and continuous curvature. over an area about four times the area of section of the jet. The stress between the fluid and plate arises chiefly in this way. The streams of fluid deviated in front of the plate, supposed for defmiteness to be movin through the fluid, receive from it forward momentum. Portions of this forward moving water are thrown off laterally at the edges of the plate, and diffused through the surrounding fluid, instead of falling to their original position behind the plate. Other portions of comparatively still water are dragged into motion to fill the space left behind the plate; and there is thus a pressure less than hydrostatic pressure at the back of the plate. The whole resistance to the motion of the plate is the sum of the excess of pressure in front and deficiency of pressure behind. This resistance is independent of any friction or viscosity in the fluid, and is due simply to its inertia resisting a sudden change of direction at the edge of the plate.

Experiments made by a whirling machine, in which the plate is fixed on a long arm and moved circularly, gave the following values of the coefficient f. The method is not free from objection, as the centrifugal force causes a flow outwards across the plate. Values of f.

Approximate V

Area of Plate

Borda. Hutton. Thibault.

V39 V24 .

0-13

0'25 I '49 1 '43 I '525

0-63

1-11

in sq. ft.

1-64 . ..

1-784

There is a steady increase of resistance with the size of the plate, in part or wholly due to centrifugal action. L. G. Dubuat (1734-1809) made experiments on a plane I ft. %1t;?re, mo;/'Sd in gflstraighté line in “Eater at 3 to 6? ft; percsecopltnil. a mg m e coe cien o excess o pressure in ron, an n e ?0efli§ :ie}ptfo{°lde f1cienC?' of pressure behind, so that f=m+n, he oun the o owing va ues:-

m=1; "=0~4s3:f=1-433-The

pressures were measured by pressure columns. Experiments by A. j. Morin (1795-1880), G. Piobert (1793-1871) and I. Didion (1798-1878) on plates of 0-3 to 2-7 sq. ft. area, drawn vertically through water, gave f=2-18; but the experiments were made in a reservoir of comparatively small depth. For similar plates moved tlgroughhaifl they (gound f =1~36, a result more in accordance with t ose w ic prece e.

For a fixed plane in a moving current of water E. Mariotte found f=1-25. Dubuat, in experiments in a current of water like those mentioned above, obtained the values m=1~186; n=o-670; f-I-856. Thibault exposed to wind pressure planes of I~I7 and 2-5 sq. ft. area, and found fto vary from 1-568 to 2-125, the mean value being f= 1-834, a result agreeing well with Dubuat. § 167. Stanton's Experiments on the Pressure inf Air on Surfaces.-At the National Physical Laboratory, London, . E. Stanton carried out a series of experiments on the distribution of pressure on surfaces in a current of air passing through an air trunk. These were on a small scale but with exceptionally accurate means of measurement. These experiments differ from those already given in that the plane is small relatively to the cross section of the current (Proc. Inst. 169 shows the distribution of pressure on a square plate. ab is the plate in Q, f vertical section. acb the distribution Q of pressure on the windward and adb that on the leeward side of the central section. Similarly aeb is the distribution of pressure on the windward and

f' afb on the leeward side of a diagonal

section. The intensity of pressure at ¢ the centre of the plate on the windward R side was in all cases p=G'u2/2g lb per

X/

Cw. Eng. clvi., 1904). Fig.

sq. ft., where G is the weight of a cubic foot of air and 11 the velocity of the current in ft. per sec. On the leeward side the negative ressure is uniform except near the ed)ges, and its value depends on the form of the plate. For a circular plate the pressure on the leeward side was 0-48 G112/2g and for a rectangular plate 0-66 CUZ/2g. For circular or square plates the resultant pressure on the plate was P=o-00126 1:2 Tb per sq. ft. where v is the velocity of the current in ft. per sec. On a lon narrow rectangular plate the resultant pressure was nearly 60% greater than on a circular plate. In later tests on larger planes in free air, Stanton found resistances 18 '}(, greater than those observed with small planes in the air trunk. § 168. Case when the Direction of Motion is obli ue to the Plane.-The determination of the pressure between a fluiclland surface in this case is of importance in many practical questions, for instance, in assigning the load due to wind ressure on sloping and curved roofs, and experiments have been mad; by Hutton, Vince, and Thibault on planes moved circularly through air and water on a whirling machine. | C

l

4 6

cl

Fig. 169.

Let AB (fig. 170) be a plane moving in the direction R making an angle ¢ with the plane. The resultant pressure between the fluid and the plane will be a normal

pressure'N. The component Rof

this normal pressure is the '~, A

resistance to the motion of the '

plane and the other component

L is a lateral force resisted by

the guides which support the

plane. Obviously


O

9 d>

R

R =N sin ¢;

L = N cos 4>. B

In the case of wind pressure on A § a sloping roof surface, R is the L N horizontal and L the vertical

component of the normal pressure.

In experiments with the whirling machine it is the resistance to motion, R, which is directly measured. Let P be the pressure on a plane moved normally through a fluid. Then, for the same plane inclined at an angle d> to its direction of motion, the resistance was found by Hutton to be

R = P(5in ¢)I'842 cos nb.

A simpler and more convenient expression given by Colonel Duchemin is

Fig. 170.

R=2P sin” ¢/(1-I-sin” ¢).

Consequently, the total pressure between the fluid and plane is N==2P sin qb/(1-1-sin' ¢>) =2P/(cosec ¢ + sin 45), and the lateral force is

L=2P sin 4> cos ¢/(1-Q-sin' d>).

In 1872 some experiments were made for the Aeronautical Society on the pressure of air on oblique planes. These plates, of I to 2 ft. square, were balanced by ingenious mechanism designed by F. H. Wenham and Spencer Browning, in such a manner that both the pressure in the direction of the air current and the lateral force were separately measured. These planes were placed opposite a blast from a fan issuing from a wooden pipe 18 in. square. The pressure of the blast varied from -1% to 1 in. 0 water pressure. The following are thebresults given in pounds per square foot of the plane, and a comparison of the experimental results with the pressures given by Duchemin's rule. hese last values are obtained by taking P =3-31, the observed pressure on a normal surface:- A 1 b Pl d D' ' R

rig eigkegzveen anelanl lrectlon 15° 20° 60° 90° Horizontal pressure R o-4 0-61 2-73 3-31 Lateral pressure L ..... I-6 1-96 I-26 . Normal pressure V L2-i-R2 1-65 2-05 3-01 3-31 Normal pressure by Duchemin's rule 1-605 2-027 3-276 3-31 WATER MoT0Rs

In every system of machinery deriving energy from a natural water-fall there exist the following parts:- 1. A supply channel or head race, leading the Water from the highest accessible level to the site of the machine. This may be an open channel of earth, masonry or wood, laid at as small a slope as is consistent with the delivery of the necessary supply of water, or it may be a. closed cast or wrought-iron pipe, laid at the natural slope of the ground, and about 3 ft. belolv the surface. In some cases part of the head race is an open channel, part; a closed pipe. The channel often starts from a small storage reservoir, constructed near the stream supplying the water motor, in which the water accumulates when the motor is not working. There are sluices or pen stocks by which the supply can be cut off when necessary.

2. Leading from the motor there is a tail race, culvert, or discharge pipe delivering the water after it has done its work at the lowest convenient level.

3. A waste channel, weir, or bye-wash is placed at the origin of the head race, by which surplus water, in floods, escapes. 4. The motor itself, of one of the kinds to be described presently, which either overcomes a useful resistance directly, as in the case of a. ram acting on a lift or crane chain, or indirectly by actuating transmissive machinery, as when a turbine drives the shafting, belting and gearing of a mill. With the motor is usually combined regulating machinery for adjusting the power and speed to the work done. This may be controlled in some cases by automatic governing machinery. § 160. Water Motors with Artijicial Sources of Energy.—The great convenience and simplicity of water motors has led to their adoption in certain cases, where no natural source of water power is available. In these cases, an artificial Source of water power is created by using a steam-engine to pump water to a reservoir at a great elevation, or to pump Water into a closed reservoir in which there is great pressure. The water flowing from the reservoir through hydraulic engines gives back the energy expended, less so much as has been wasted by friction. Such arrangements are most useful where a continuously acting steam engine stores up energy by pumping the water, while the work done by the hydraulic engines is done intermittently.

§ 170. Energy of a U/ater-fall.-Let H; be the total fall of level from the point where the water is taken from a natural stream to the point where it is discharged into it again. Of this total fall a portion, which can be estimated independently, is expended in overcoming the resistances of the head and tail races or the supply and discharge pipes. Let this portion of head wasted be i), . Then the available head to work the motor is H = H, -br. It is this available head which should be used in all calculations of the proportions of the motor. Let Q be the supply of water per second. Then GQH foot-pounds per second is the gross available work of the fall. The power of the fall may be utilized in three ways. (zz) The GQ pounds of water may be placed on a machine at the highest level, and descending in contact with it a distance of H ft., the work done will be (neglecting losses from friction or leakage) GQH foot-pounds per second. (b) Or the water may descend in a closed pipe from the higher to the lower level, in which case, with the same reservation as before, the pressure at the foot of the pipe will be p = GH pounds per square foot. If the water with this pressure acts on a movable piston like that of a steam engine, it will drive the piston so that the volume described is Q cubic feet per second. Then the work done will be pQ=GHQ foot-pounds per second as before. (c) Or lastly, the water may be allowed to acquire the velocity v= V zgl-i by its descent. The kinetic energy of Q cubic feet will then be § GQv2/g=GQH, and if the water is allowed to impinge on surfaces suitably curved which bring it finally to rest, it will impart to these the same energy as in the previous cases. Motors which receive energy mainly in the three WZIYS C1€SCYibCd ifl (fl). (17), (6) may be termed gravity, pressure and inertia motors respectively. Generally, if Q ft. per second of water act by weight through a distance hl, at a pressure p due to hg ft. of fall, and with a velocity 'v due to h3 ft. of fall, so that hi-l-hz-l-h3=H, then, apart from energy wasted by friction or leakage or imperfection of the machine, the work done will be

GQh1-l-pQ-l-(G/g)Q(v2/zg) =GQH foot pounds,

the same as if the water acted simply by its weight while descending H ft.

§ 171. Site for Water Motor.-Wherever a stream flows from a higher to a lower level it is possible to erect a water motor. The amount of power obtainable depends on the available head and the supply of water. In choosing a site the engineer will select a portion of the stream where there is an abrupt natural fall, or at least a considerable slope of the bed. He will have regard to the facility of constructing the channels which are to convey the water, and will take advantage of any bend in the river which enables him to shorten them. He will have accurate measurements made of the quantity of water flowing in the stream, and he will endeavour to ascertain the average quantity available throughout the year, the minimum quantity in dry seasons, and the maximum for which bye-wash channels must be provided. In many cases the natural fall can be increased by a dam or weir thrown across the stream. The engineer will also examine to what extent the head will vary in different seasons, and whether it is necessary to sacrifice part of the fall and give a steep slope to the tail race to prevent the motor being drowned by backwater in floods. Streams fed from lakes which form natural reservoirs or fed from glaciers are less variable than streams depending directly on rainfall, and are therefore advantageous for water-power purposes.

§ 172. Water Power at Holyoke, U.S.A.-About 85 m. from the mouth of the Connecticut river there was a fall of about 60 ft. in a short distance, forming what were called the Grand Rapids, below which the river turned sharply, forming a kind of peninsula on which the city of Holyoke is built. In 1845 the magnitude of the water power available attracted attention, and it was decided to build a dam across the river. The ordinary How of the river is 6000 cub. ft. per sec., giving a gross power of 30.000 h.p. In dry seasons the power is 20,000 h.p., or occasionally less. From above the dam a system of canals takes the water to mills on three levels. The first canal starts with a width of 140 ft. and depth of 22 ft., and supplies the highest range of mills. A second canal takes the water which has driven turbines in the highest mills and supplies it to a second series of mills. There is a third canal on a still lower level supplying the lowest mills. The Water then finds its way back to the river. /Vith the grant of a mill site is also lcased the right to use the water power. A mill-power is defined as 38 cub. ft. of water per sec. during 16 hours per day on a fall of 20 ft. This gives about 60 h.p. effective. The charge for the power water is at the rate of 205. per h.p. per annum.

§ 173. Action of Water in a WolerMolor.-Water motors may be divided into water-pressure engines, water-wheels and turbines.

Water-pressure engines are machines with a cylinder and piston or ram, in principle identical with the corresponding part of a steam-engine. The water is alternately admitted to and discharged from the cylinder, causing a reciprocating action oi the piston or plunger. It is admitted at a high pressure and discharged at a low one, and consequently work is done on the piston. The water in these machines never acquires a high velocity, and for the most part the kinetic energy of the water is wasted. The useful work is due to the difference of the pressure of admission and discharge, whether that pressure is due to the weight of a column of water of more or less considerable height, or is artificially produced in ways to be described presently. Water-wheels are large vertical wheels driven by water falling from a higher to a lower level. In most Water-wheels, the water acts directly by its weight loading one side of the wheel and so causing rotation. But in all water-wheels a portion, and in some a considerable portion, of the work due to gravity is first employed to generate kinetic energy in the water; during its action on the water-wheel the velocity of the water diminishes, and the wheel is therefore in part driven by the impulse due to the change of the water's momentum. Water-wheels are therefore motors on which the water acts, partly by weight, partly by impulse.

Turbines are wheels, generally of small size compared with water wheels, driven chiefly by the impulse of the water. Before entering the moving part of the turbine, the water is allowed to acquire a considerable velocity; during its action on the turbine this velocity is diminished, and the impulse due to the change of momentum drives the turbine.

In designing or selecting a water motor it is not sufficient to consider only its efficiency in normal conditions of working. It is generally quite as important to know how it will act with a scanty water supply or a diminished head. The greatest difference in water motors is in their adaptability to varying conditions of working.

Water-pressure Engines.

§ 174. In these the water acts by pressure either due to the height of the column in a supply pipe descending from a high level reservoir, or created by pumping. Pressure engines were first used in mine-pumping on waterfalls of greater height than could at that time be utilized by water wheels. Usually they were single acting, the water-pressure lifting the heavy pump rods which then made the return or pumping stroke by their own weight. To avoid losses by fluid friction and shock the velocity of the water in the pipes and passages was restricted to from 3 to ro ft. per second, and the mean speed of plunger to 1 ft. per second. The stroke was long and the number of strokes 3 to 6 per minute. The pumping lift being constant, such engines worked practically always at full load, and the efficiency was high, about 84%. But they were cumbrous machines. They are described in Weisbach's M echonics of Engirieering. The convenience of distributing energy from a central station to scattered working-points by pressure water conveyed in pipes -a system invented by Lord Armstrong-has already been mentioned. This system has led to the development of a great variety of hydraulic pressure engines of very various types. The cost of pumping the pressure water to some extent restricts its use to intermittent operations, such as working lifts and cranes, punching, shearing and riveting machines, forging and flanging presses. To keep down the cost of the distributing mains very high pressures are adopted, generally 700 lb per sq. in. or 1600 ft. of head or more.

In a large number of hydraulic machines worked by water at high pressure, especially lifting machines, the motor consists of a direct, single acting ram and cylinder. In a few cases double acting pistons and cylinders are used; but they involve a water-tight packing of the piston not easily accessible. In some cases pressure engines are used to obtain rotative movement, and then two double-acting cylinders or three single-acting cylinders are used, driving a crank shaft. Some doub1e»acting cylinders have a piston rod half the area of the piston. The pressure water acts continuously on the annular area in front of the piston. During the forward stroke the pressure on the front of the piston balances half the pressure on the back. During the return stroke the pressure on the front is unopposed. The water in front of the piston is not exhausted, but returns to the supply pipe. As the frictional losses in a fluid are independent of the pressure, and the work done increases directly as the pressure, the percentage loss decreases for given velocities of flow as the pressure increases. Hence for high-pressure machines somewhat greater velocities are permitted in the passages than for low-pressure machines. In supply mains the velocity is from 3 to 6 ft. per second, in valve passages 5 to ro ft. per second, or in extreme cases zo ft. per second, where there is less object in economizing energy. As the water is incompressible, slide valves must have neither lap nor lead, and piston valves are preferable to ordinary slide valves. To prevent injurious compression from exhaust valves closing too soon in rotative engines with a fixed stroke, small self-acting relief valves are fitted to the cylinder ends, opening outwards against the pressure into the valve chest. Imprisoned water can then escape without over straining the machines.

In direct single-acting lift machines, in which the stroke is fixed, and in rotative machines at constant speed it is obvious that the cylinder must be filled at each stroke irrespective of the amount of work t.o be done. The same amount of water is used whether much or little work is done, or whether great or small weights are lifted. Hence while pressure engines are very efficient at full load, their efficiency decreases as the load decreases. Various arrangements have been adopted to diminish this defect in engines working with a variable load. In lifting machinery there is sometimes a double ram, a hollow ram enclosing a solid ram. By simple arrangements the solid ram only is used for small loads, but for large loads the hollow ram is locked to the solid ram, and the two act as a ram of larger area. In rotative engines the case is more difficult. In Hastie's and Rigg's engines the stroke is automatically varied with the load, increasing when the load is large and decreasing when it is small. But such engines are complicated and have not achieved much success. Where pressure engines are used simplicity is generally a first consideration, and economy is of less importance. § 175. Ejiciency of Pressure Engines.—It is hardly possible to form a theoretical expression for the efficiency of pressure engines, but some general considerations are useful. Consider the case of a long stroke hydraulic ram, which has a fairly constant velocity v during the stroke, and valves which are fairly wide open during most of the stroke. Let r be the ratio of area of ram to area of valve passage, a ratio which may vary in ordinary cases from 4 to 12. Then the loss in shock of the water entering the cylinder will be (r-I)”'v2/2g in ft. of head. The friction in the supply pipe is also proportional to rf. The energy carried away in exhaust will be proportional to 112. Hence the total hydraulic losses may be taken to be approximately tu?/2g ft., where § ' is a coefficient depending on the proportions of the machine. Let f be the friction of the ram packin and mechanism reckoned in Tb per sq. ft. of ram area. Then ig the supply-pipe pressure driving the machine is p lb per sq. ft., the effective working pressure will be

p-C5112/2g-f lb per sq. ft.

Let A be the area of the ram in sq. ft., v its velocity in ft. pa' sec. The useful work done will be

(p-Cya'/ag-f)At' ft. Tb per sec.,

and the efficiency of the machine will be 11 = (P 'G?1'2/22'°f)/P-This

shows that the efficiency increases with the pressure p, and diminishes with the speed v, other things being the same. If in regulating the engine for varying load the pressure is throttled, part of the available head is destroyed at the throttle valve, and p in the bracket above is reduced. without intermediate gearing, may

have an efficiency of 95 % during the working stroke. If a hydraulic jigger is used with roipes and sheaves to change Direct-acting hydraulic lifts,

4

the speed o the ram to the speed of the lift, the efficiency may be only 1 .

/

f

Level of Supply

I

50 %. E. B. Ellington has given the Efficiency t cg Jiftis witlgh hydraulic, V 3 HHCC 3 Uflfl 6 WOT 1I'l

stroke. Large pressurg engines havi a n efficiency of 85 %, but small rota- ; r / tive engines probably not more than I, i Z I 50 % and that only when fully loaded.:fi I 1 Z } 512 WE 3 /

§ 176. Direct-Acting Hydraulic, QQ: Z Lift (fig. 17r>. This is the Z f simplest of all kinds of hydraulic I Z § motor. A cage W is lifted directly I ' / 2 by water pressure acting in a Z:

cylinder C, the length of which is Z I a little greater than the lift. A ' Z ram or plunger R of the same l Z

length is attached to the cage. I The water-pressure admitted by a Z Q cock to the cylinder forces up the 'I ' Z: ram, and when the supply valve is H Z I closed and the discharge valve 'i ' Z: opened, the ram descends. In 5 » this case the ram is 9 in. diameter, 1 » Z I with a stroke of 49 ft. It consists Z § of lengths of wrought-iron pipe I ° Z } screwed together perfectly water- —% Q tight, the lower end being closed r %: by a. cast-iron plug. The ram % »

works in a cylinder II in. dia- rg % meter of 9 ft. lengths of Banged % cast-iron pipe. The ram passes, I Z % water-tight through the cylinder E, % I cover, which is provided with %

double hat leathers to prevent I: % If leakage outwards or inwards. As E, % I the weight of the ram and ca ' 5: - l% 1 ge is E %:

much more than sufficient to cause B, , N % a descent of the cage, part of the UFQ % weight is balanced. A chain at- % I tached to the cage passes over a ' 7 "Z % }“ pulley at the top of Z:

the lift, and carries 1% AW, n

at its free end a L A' 'E' ' 1

balance weight B, Disgymrge :

working in T iron i Q, 4 4* I

guides. Water is ad-:, , W :

mitted to the cylinder F it:N

from a 4-in. supply i Y, Ee ,

pipe through a two- 2. 1 2

way slide, worked by { C, Q 1

a rack, spindle and: 2.endless rope. The:

lift works under 73 Hb f r E ” 1 Q ft. of head, and lifts i 3/ Q

1350 Tb at 2 ft. per l 5* Q

second. The efh- } f iciency

is from 75 to 2 s

80%.: 2 /1 1

ll ° ' - ' :ff

jucrhciial pfeggghceprfto i, =

the motion of a ram I fi, "Iliff

of this kind is the fric- ! ...-..— / f. .'; .. 'C. tion of the cup leathers, '; wévhich makge therjoént

8 I1

anilweifhmf eSoC1iidne;€ FIG' 171° I pediments by John Hick give for the friction of these leathers the following formula. Let F= the total friction in pounds; d=diameter of ram in ft.; p=water-pressure in pounds per sq. ft.; k a coefficient. F=kpd k=O~OO393 if the leathers are new or badly lubricated; =o-00262 if the leathers are in good condition and well lubricated. Since the total pressure on the ram is P==i1rd2p, the fraction of the total pressure expended in overcoming the friction of the leathers is F/P= -oo5/d to -0033/d, d being in feet. Let H be the height of the pressure column measured from the free surface of the supply reservoir to the bottom of the ram in its lowest position, Hy, the height from the discharge reservoir to the same point, h the height of the ram above its lowest point at any moment, S the length of stroke, S2 the area of the ram, W the weight of cage, R the weight of ram, B the weight of balance weight, 'w the weight of balance chain per foot run, F the friction of the cup leather and slides. Then, neglecting fluid friction, if the ram is rising the accelerating force is P, =G(H-h)SZ-R-W+B-'w(S-h) -I-wh-F, and if the ram is descending P2= -G(H;, -h)SZ+W-I-R-B+'w(S-h)-wh-F. If ~w=% GS), P1 and P2 are constant throughout the stroke; and the moving force in ascending and descending is the same, if B =W-I-R+wS-Gf2(l-I-l-Hb)/2. Using the values just found for 'w and B, P1 = P2 = § G§ Z(H-Hb) -F. Let W+R+wS+B=U, and let P be the constant accelerating force acting on the system, then the acceleration is (P/U)g The velocity at the end of the stroke is (assuming the friction to be constant) v=~J (2P€5/U): and the mean velocity of ascent is év. § 177. Armstrong? Hydraulic J igger.-This is simply a single acting hydraulic cylinder and ram, provided with sheaves so as to give motion to a wire rope or chain. It is used in various forms of lift and crane. Fig. 172 shows the arrangement. A hydraulic ram or plunger B works in a stationary cylinder A. Ram and cylinder ®< carry sets of sheaves over which passes a chain or rope, fixed at one end to the / cylinder, and at the other connected over @ ”

(°/ guide pulleys to a lift or crane. For each pair of pulleys, one on the cylinder and one on the ram, the movement of the free end of the rope is doubled compared with that of the ram. With three pairs of pulleys the free end of the rope has a movement equal to six times the stroke of the ram, the force exerted being in the inverse proportion. - § 178. Rotalive Hydraulic Engines.-Valvegear mechanism similar in principle to that of steam engines can be applied to actuate the admission and discharge valves, and the pressure engine is then converted into a continuously-acting motor.

k, Let H be the available fall to work the engine after deducting the loss of head in the supply and discharge pipes, Q the supply of water in cubic feet per second, and 1; the efficiency of the engine. Then the horse-power of the en me is 3 H.P.=1;GQH/550. The efficiency of large slow-moving pressure engines is 17= '66 to -8. In small motors of this kind probably 17 IS not greater than -5. Let 'U be the mean velocity of the piston, then its diameter d is given by the relation Q= vrdzv/4 in double-acting engines, =1rd2'v/8 in single-acting engines. If there are n cylinders put Q/n for Q in these equations. Small rotative pressure engines form extremely convenient motors for hoists, capstans or Winches, and for driving small machinery. The single-acting engine has the advantage that the pressure of the piston on the crank pin is always in one direction; there is then no knocking as the dead centres are passed. Generally three single-acting cylinders are used, so that the engine will readily start in all positions, and the driving effort on the crank pin is very uniform. K/ FIG. 172. Brotherhood Hydraulic Engine.-Three cylinders at angles of 120° with each other are formed in one casting with the frame. The plungers are hollow = trunks, and the connecting rods abut in cylindrical recesses in them and are connected to a common crank pin. A circular valve disk with concentric segmental ports revolves at the same rate as the crank over ports in the valve face common to the three cylinders. Each cylinder is always in communication with either an admission or exhaust port. The blank parts of the circular valve close the admission and exhaust ports alternately. The fixed valve face is of li num vitae in a metal recess, and the revolving valye of gun-meta? In the case of a small capstan engine the the piston speed is 31 ft. per minute. The ports are I in. diameter orilg of the piston area, and the mean velocity in the ports 6-4 ft. per sec. With 700 Tb per sq. in. water pressure and an efficiency of 50%, the engine is about 3 h.p. A common arrangement is to have three parallel cylinders acting on a three-throw crank shaft, the cylinders oscillating on trunnions. Hasliels Engine.-Fig. 173 shows a similar engine made by Messrs Hastie of Greenock. G, G, G are the three cylinders are 3% in. diameter and 3 in. stroke. At 40 revs. per minute. E A s G7-'W /, aw J“ it If,) " ' W 'ann // ° '%(/M Fig. 173. plungers which pass out 7 of the cylmderstlirough cup leathers, and act on the same crank pin A is the 1nl t l ' ° e pipe wiich communicates with the cock B. This cock controls the action of the engine, being so constructed that it acts as a reversing valve when the handle C is in its extreme ositions and as a brake when 111 its middle position. ¢Vith the fiandle in its m}dd1¢ IJ0S1U0I1, the ports of the cylinders are in communication w1th the exhaust. Two passages are formed in the framing 'leading from the cock B to the ends of the cylinders, one being in communication with the supply pipe A, the other with the discharge pipe Q. These passages cylinders puts them end as shown at E. The oscillation of the alternately in communication with each of @ , these passages, and thus 5 the water IS alternately -', / 1 K admitted and exhausted @ 3 In any ordinary rota- ', tive engine the length of 'G ff stroke is invariable. Z6 — Consequentlty the con- sumption o water depends simply on the speed of the engine, irrespective of the effort overcome. If the power of the engine must be varied without altering the number of rotations, then the stroke must be made variable. Messrs Hastie have contrived an exceedingly ingenious method of varying the stroke FIG. 174. 174). The crank(pin is carried in a sli e H automatically, in proportion to the amount of work to be done (fig. f""' ~ ' r moving in a disk M., fag, In this IS a double 1 § cam K acting on two E* »'O'; 3 g small steel rollers ], 3, .' '. -', L attached to the 4 5 slide H. If the Cam ., r~»<° T Q rotates it moves the 8 E slide and increases or 5 N 2 H»~ N ¢ E decreases the radius of Q il (7/'Ly S the circle in which the ' ll, QQ) gg crank pin I rotates. § 1', W »< The disk M is keyed T s if, on a hollow shaft sur- T 11, T . H rounding the driving 7' 'E .j - R 'Q shaft P, to which the ' '§ i »cams are attached. f i 13 €7 Elie hollow shaft N § § °" I as two snugs to ~ . § . ., .» which the chains RR "'a;>r g A/fs" are attached (fig. 175). The shaft Pcarries the A F G springcaseSStowhich I ' 175 also are attached the other ends of the chains. When the engine is at rest the springs extend themselves, rotating the hollow shaft N and the frame M, so as to place the crank pin I at its nearest position to the axis of rotation. W'hen a resistance has to be overcome, the shaft N rotates relatively to P, compressing the springs, till their resistance balances the pressure due to the resistance to the rotation of P. The engine then commences to work, the crank pin being in the position in which the turning effort just overcomes the resistance. If the resistance diminishes, the springs force out the chains and shorten the stroke of the plungers, and vice versa. The following experiments, on an engine of this kind working a hoist, show how the automatic arrangement adjusted the water used to the work done. The lift was 22 lt. and the water pressure in the cylinders 80 lb per sq. in. W. . § , § g'§ f"“ed' EE, " 427 633 745 S51 969 1081 1193 W;§ '&;Eed'i“ 7% 10 14 16 17 20 21 22 § 179. /lCClH7l1¢Z<1Z0l' M achinery.-It has already been pointed out that it is in some cases convenient to use a steam engine to create an artificial head of water, which is afterwards employed in driving water-pressure machinery. Where power is required intermittently, for short periods, at a number of different points, as, for instance, in moving the cranes, lock gates, &c., of a dockyard, a separate steam engine and boiler at each point is very inconvenient; nor can engines worked from a common boiler be used, because of the great loss of heat and the difficulties which arise out of condensation in the pipes. If a tank, into which water is continuously pumped, can be placed at a great elevation, the water can then be used in hydraulic machinery in a very convenient way. Each hydraulic machine is put in communication with the tank by a pipe, and on opening a valve it commences work, using a quantity of water directly proportional to the work done. No attendance is required when the machine is not working. A site for such an elevated tank is, however, seldom available, and in place of it a beautiful arrangement termed an accumulator, invented by Lord Armstrong, is used. This consists of a tall vertical cylinder; into this works a solid ram through cup leathers or hemp packing, and the ram is loaded by fixed weights, so that the pressure in the cylinder is 700 lb or Soo lb er sq.in ing a cylindrical wrought-iron tank W, in which weights are placed to load the accumulator. At R is one of the pressure engines or j iggers, worked from the accumulator, discharging the water after use into the tank T. In this case the pressure engine is shown working a set of blocks, the fixed block being on the ram cylinder, the running block on the ram. The chain running over these blocks works a lift cage C, the speed of which is as many times greater than that of the ram as there are plies of chain on the block tackle. B is the balance 5 weight of the cage. In the use of accumulators on shipboard for working gun gear or steering gear, the accumulator ram is loaded by f springs, or by steam pressure acting on a 2 piston much larger than the ram. 2 R. H. Tweddell has used accumulators with a pressure of 2000 lb per A 9 stqiniéiéyto work hydraulic riveting ma- lf? 3C;.h.iui§ l§ ?, “ti.§ i1§ '§ e'§ y.§§ .'§ 'e§ i'$.“i§ ?;;f2§ t;'i.ir;;';:.°;.§ ;;: § .'fd.;'€'”@f¢ ZpdYS foot-pounds., 2 Thus, if the ram is 9 in., the stroke 20 ft., and the pressure 800 lb per sq. in., the work stored in the accumulator when the Q ram is at the top of the stroke is 1,01 7,600 1% foot-pounds, that is, enough to drive a machine requiring one horse power for '“ Z f& /® f

Eg E U > ze Q

.

l

§ /, . ” /il &/ / /

7 about half an hour. As, however, the > li pumping engine replaces water as soon / ll as it is drawn off, the working capacity ii, A of the accumulator is very much greater amd Q; Md than this. Tweddell found that an ac- sit;, iicumulaticg charged at E50 lb dqisqharged W at 1225 pers .in. - ence the riction was equivalentiio I2% lb per sq. in. and the efficiency 98 %. ' ” '”" ' " Vl/'hen a very great pressure is required FIG 177. p . In some cases the ram is fixed and the cylinder moves on it. e>t"e>%ff 1 | I | i.

1 1 C I | Q <, r 1 I if - - l ll Z i fl f l-i I 3 ¢,

J

1, :, V 7° . ~, , pi R .i ~ 17- E 1 =' -, ll "'*"" . [1 N;,

., ' ' I

L, - L-, ll / i, |. ... V, ,, ,, ,1FfHf7"'5f!77'ffF?F1f17ffZ' 57 l7'#ff» " / ' / / ww/mn/» f wi / I 1////7/I/I1////////////11/1' I/ " "//" FIG. 176. . The pumping engin es which supply the energy that is stored in the accumulator should be a pair coupled at right angles, so as to start in any position. The engines pump into the accumulator cylinder till the ram is at the top of its stroke, when by a catch arrangement acting on the engine throttle valve the engines are stopped. If the accumulator ram descends, in consequence of water being taken to work machinery, the engines immediately recom- mence working. Pipes lead from the accumulator to each of the machines requiring to be driven, and do not require to be of large size, as the pressure is so great Fig. 176 shows a diagrammatic way the scheme of a system of accumulator machinery. A is the accumulator, with its ram carry a differential accumulator (fig. 177) is convenient. The ram is fixed and passes through both ends of the cylinder, but is of different diameters at the two ends, A and B. Hence if di, d2 are the diameters of the ram ininches and p the required pressure in lb per sq. in., the load required is }p1r(d”, -dk). An accumulator of this kind used with riveting machines has di = 5% in., dz =4gll'1. The pressure is 2000 lb per sq. in. and the load 5~4 tons. Sometimes an accumulator is loaded by water or steam pressure instead of by a dead weight. Fig. 178 shows the arrangement. A piston A is connected to a plunger B of much smaller area. Vater pressure, say from town mains, is admitted below A, and the high '“" pressure water is pumped into and discharged from the cylinder C in which B works. If r is the ratio of the areas of A and B, then, neglecting friction, the pressure in the upper cylinder is 1' times that under the piston A. Vllith a variable rate of supply and demand from the upper cylinder, the piston A rises and fa1ls, maintaining always a constant pressure in the upper cylinder. /M/////m/n 0 //////////y / /4//0 ///// / 0////4 1 0u£[e[ i I qi %//, 1A 0/0//5 , ls gx %%/ My/pa////M 4

°~ $ 3

@ I mzffaz
- W///////n / /

4 4

N

4/1/////////an///////////////m//////0 § ?§ Water W heels. § 180. Overshot and High Breast Wheels. -When a water fall ranges between IO and 70 ft. and the water supply is from 3 to 25 cub. ft. per second, it is possible to construct a bucket wheel on which the water acts chiefly by its weight. If the variation of the head-water level does not exceed 2 ft., an overshot wheel may be used (fig. 179). The water is then projected over the summit of the wheel, and falls in a parabolic path into the buckets. With greater variation of head-water level, a pitch-back or high breast Wheel is better. The water falls over the top of a sliding sluice into the wheel, on the same side as the head race channel. By adjusting the height of the sluice, the requisite supply is given to the wheel in all positions of the head-water level. The wheel consists of a cast-iron or wrought-iron axle C supporting the weight of the wheel. To this are attached two

W /' 1 Fig. 178. sets of arms A of wood or iron, which support circular segmental plates, B, termed shrouds. A cylindrical sole plate dd extends between the shrouds on the inner side. The buckets are formed

Frc. 179. by wood planks or curved wrought-iron plates extending from shroud to shroud, the back of the buckets being formed by the sole plate. The efhciency may be taken at 0-75. Hence, if h.p. is the effective horse power, H the available fall, and Q the available water supply per second, h. p. =o-75(GQH/550)=o-085 QH. If the peripheral velocity of the water wheel is too great, water is thrown out of the buckets before reaching the bottom of the fall. In practice, the circumferential velocity of water wheels of the kind now described is from 4% to IO ft. per second, about 6 ft. being the usual velocity of good iron wheels not of very small size. In order that the water may enter the buckets easily, it must have a greater velocity than the wheel. Usually the velocity of the water at the point where it enters the wheel is from 9 to 12 ft. per second, and to produce this it must enter -the wheel at a point 16 to 27 in. below the head-water level. Hence the diameter of an overshot wheel may be D=H-1% to H-2% ft. Overshot and high breast wheels work badly in back-water, and hence if the tail-water level varies, it is better to reduce the diameter of the wheel so that its greatest immersion in flood is not more than I ft. The depth d of the shrouds is about IO to 16 in. The number N = 1rD/ti. Let v be the peripheral velocity of the wheel. Then the capacity of that portion of the wheel which passes the sluice in one second is Q1=vb(Dd-d2)/D =v b d nearly, b being the breadth of the wheel between the shrouds. If, however, this quantity of water were allowed to pass on to the wheel the buckets would begin to spill their contents almost at the top of the fall. To diminish the loss from spilling, it is not only necessary to give the buckets a suitable form, but to restrict the water supply to one-fourth or one-third of the gross bucket capacity. Let m be the value of this ratio; then, Q being the supply of water per second, Q=mQ1=mbdv. This gives the breadth of the wheel if the water supply is known. The form of the buckets should be determined thus. The outer element of the bucket should be in the direction of motion of the water entering relatively to the wheel, so that the water may enter without splashing or shock. The buckets should retain the water as long as possible, and the width of opening of the buckets should be 2 or 3 in. greater than the thickness of the sheet of water entering. of buckets may be about join cd. For an iron b Draw 50 making an uck G11 For a wooden bucket (fig. 180, A), take ab=distance between two buckets on periphery of wheel. Make ed=% eb. and bc=§ to 2 ab. (fig. 180, B), take ed=% eb; bc= gab. angle of 10° to 15° with g A B the radius at c. Qn Oc ' § take a centre giving a cg § ~. c circular arc passing g, ...»- " "g"Tf§ f, § , § ff near d, and round the § ;.i<—r curve into the radial § "" part of the bucket zie. e There are two ways ~, in which the power of I d 5 “ge ', a water wheel is given sr "" 5 ~ oft' to the machinery §

driven. In wooden;< § Q wheels and wheels ~~-3 al with rigid arms, a spur ' Slowking4Farmbrough's revenge 02:27, 19 April 2015 (UTC) 2 Q " " or bevil wheel keyed ' s on the axle of the F 8 turbine will transmit IG' I O the power to the shafting. It is obvious that the whole turning moment due to the weight of the water is then transmitted through the arms and axle of the water wheel. When the water wheel is an iron one, it usually has light iron suspension arms incapable of resisting the bending action due to the transmission of the turning effort tc the axle. In that case spur segments are bolted to one of the shrouds, and the pinion to which the power is transmitted is placed so that the teeth in gear are, as nearly as may be, on the line of action of the resultant of the weight of the water in the loaded arc of the wheel. The largest high breast wheels ever constructed were probably the four wheels, each 50 ft. in diameter, and of 125 h.p., erected by Sir W. Fairbairn in 182 5 at Catrine in Ayrshire. These wheels are still working. § 181. Poncelet W aler Wheel.-When the fall does not exceed 6 ft., the best water motor to adopt in many cases is the Poncelet undershot Water wheel. In this the water acts very nearly in the same way as in a turbine, and the Poncelet wheel, although slightly less ethcient than the best turbines, in normal conditions of working, is superior to most of them when working with a reduced supply of water. A general notion of the action of the water on a Poncelet wheel has already been given in § 159. Fig. 181 shows its construction. The water penned back between the side walls of the wheel pit is allowed to flow to the | 2 'I } 1 - A tl oil/

3, V I.:- .1: T °;ia 1 f, f 3 Q, ~ H ~-~—eo; cc . l, , - — - 1 ' » », ¢ » Z Q; J. , . I #WW/T E -<»i'* "wi;' '4»@, ' Y* ° l W“*'<" " T”/ f iffif-*if E W / »// '-»“ “' * - -~~- '- 1 1 'v " -' *~ ~ - - -'~ ~//,

  • " * "' *'-I

4-I /Wa/, » Fig. 181. wheel under a movable sluice, at a Velocity nearly equal to the velocity due to the whole fall. The water is guided down a slope of 1 in 10, or a curved race, and enters the wheel without shock. Gliding up the curved floats it comes to rest, falls back, and acquires at the point of discharge a backward velocity relative to the wheel nearly equal to the forward velocity of the wheel. Consequently it leaves the wheel deprived of nearly the whole of its original kinetic energy. Taking the efficiency at 0-60, and putting H for the available fall, h.p. for the horse-power, and Q for the water supply per second, h.p. = 0-068 QH. The diameter D of the wheelmay be taken arbitrarily. It should not be less than twice the fall and is more often four times the fall. For ordinary cases the smallest convenient diameter is 14 ft. with a straight, or IO ft. with a curved, approach channel. The radial depth of bucket should be at least half the fall, and radius of curvature of buckets about half the radius of the wheel. The shrouds are usually of cast iron with flanges to receive the buckets. The buckets may be of iron § in thick bolted to the flanges with 156 in. bolts. Let H' be the fall measured from the free surface of the head water to the point F where the mean layer enters the wheel; then the velocity at which the water enters is v=~/(2gH'), and the best circumferential velocity of the wheel is V=0~55v to o-611. The number of rotations of the wheel per second is N=V/1rD. The thickness of the sheet of water entering the wheel is very important. The best thickness according to experiment is 8 to IO in. The maximum thickness should not exceed 12 to 15 in., when there is a surplus water supply. Let e be the thickness of the sheet of water entering the wheel, and b its width; then bev=Q; or b=Q/ev.

Grashof takes e=§ -H, and then

b =6Q/H sf (2£H)-Allowing

for the contraction of the stream, the area of opening through the sluice may be 1-25 be to 1-3 be. The inside width of the wheel is made about 4 in. greater than b. Several constructions have been given for the floats of Poncelet wheels. One of the simplest is that shown in figs. 181, 182. Let OA (fig. 181) be the vertical radius of the wheel. Set off OB, OD making angles of 15° with OA. Then BD may be the length of ~

1

r

io

fi

l, v

K:—~-. .i— K?

= aff 1

"/I -/iii ' i;} x;

-ffffi/Lim/Z&£Mf.WLWW'f' 1 -' 'l . 'u“ ~ f .»-' o

av

~

1 v, .

G

Fig. 182.

the close breasting fitted to the wheel. Draw the bottom of the head race BC at a slope of I in 10. Parallel to this, at distances ée and e, draw EF and GH. Then EF is the mean layer and GH the surface layer entering the wheel. ]oin OF, and make OFK=23°. Take FK=o-5 to o-7 H. Then K is the centre from which the bucket curve is struck and KF is the radius. The depth of the shrouds must be sufficient to prevent the water from rising over the top of the float. It is § H to § H. The number of buckets is not very important. They are usually 1 ft. apart on the circumference of the wheel.

The efficiency of a Poncelet wheel has been found in experiments to reach 0-68. It is better to take it at 0-6 in estimating the power of the wheel, so as to allow some margin. In fig. 182 fs; is the initial and v., the final velocity of the water, v, parallel to the vane the relative velocity of the water and wheel, and V the velocity of the wheel.

T urbines.§

182. The name turbine was originally given in-France to any water motor which revolved in a horizontal plane, the axis being vertical. The rapid development of this class of motors dates from 1827, when a prize was offered by the Société d'Encouragement for a motor of this kind, which should be an improvement on certain wheels then in use. The prize was ultimately awarded to Benoit Fourneyron (1802-1867), whose turbine, but little modified, is still constructed. Classification of Turbines:-In some turbines the whole available energy of the water is converted into kinetic energy before the water acts on the moving part of the turbine. Such turbines are termed Impulse or Action Turbines, and they are distinguished by this that the wheel passages are never entirely filled by the water. To ensure this condition they must be placed a little above the tail water and discharge into free air. Turbines in which part only of the available energy is converted into kinetic energy before the water enters the wheel are termed Pressure or Reaction Turbines. In these there is a pressure which in some cases amounts to half the head in the clearance space between the guide vanes and wheel vanes. The velocity with which the water enters the wheel is due to the difference between the pressure due to the head and the pressure in the clearance space. In pressure turbines the wheel passages must be continuously filled with water for good etiiciency, and the wheel may be and generally is placed below the tail water level. Some turbines are designed to act normally as impulse turbines discharging above the tail water level. But the passages are so designed that they are just filled by the water. If the tail water rises and drowns the turbine they become pressure turbines with a small clearance pressure, but the efficiency is not much affected. Such turbines are termed Limit turbines. Next there is a difference of constructive arrangement of turbines, which does not very essentially alter the mode of action of the water. In axial flow or so-called parallel flow turbines, the water enters and leaves the turbine in a direction parallel to the axis of rotation, and the paths of the molecules lie on cylindrical surfaces concentric with that axis. In radial outward and inward How turbines, the water enters and leaves the turbine in directions normal to the axis of rotation, and the paths of the molecules lie exactly or nearly in planes normal to the axis of rotation. In outward flow turbines the general direction of flow is away from the axis, and in inward How turbines towards the axis. There are also mixed How turbines in which the water enters normally and is discharged parallel to the axis of rotation. Another difference of construction is this, that the water may be admitted equally to every part of the circumference of the turbine wheel or to a portion of the circumference only. In the former case, the condition of the wheel passages is always the same; they receive water equally in all positions during rotation. In the latter case, they receive water during a part of the rotation only. The former may be termed turbines with complete admission, the latter turbines with partial admission. A reaction turbine should always have complete admission. An impulse turbine may have complete or partial admission. When two turbine wheels similarly constructed are placed on the same axis, in order to balance the pressures and diminish journal friction, the arrangement may be termed a twin turbine. If the water, having acted on one turbine wheel, is then passed through a second on the same axis, the arrangement may be termed a compound turbine. The object of such an arrangement would be to diminish the speed of rotation. Many forms of reaction turbine may be placed at any height not exceeding go ft. above the tail water. They then discharge into an air-tight suction pipe. The weight of the column of water in this pipe balances part of the atmospheric pressure, and the difference of pressure, producing the flow through the turbine, is the same as if the turbine were placed at the bottom of the fall. II. Reaction Turbines.

(Wheel passages filled, discharging above or below the tail

water or into a suction-pipe.)

Always with complete admission.

v,

I. Impulse Turbines.

(Wheel passages not filled, and

discharging above the tail

water.)

(a) Completeadmission. (Rare.)

(b) Partial admission. (Usual.)

Axial How, outward flow, inward How, or mixed How.

Y

Simple turbines; twin turbines; compound turbines. § 183. The Simple Reaction Wheel.-It has been shown, in § 162, that, when water issues from a vessel, there is a reaction on the vessel tending to cause motion in a direction opposite to that of the jet. This principle was applied in a rotating water motor at a very early period, and the Scotch turbine, at one time much used, diliers in no essential respect from the older form of reaction wheel.

The old reaction wheel consisted of a vertical pipe balanced on a vertical axis, and supplied with water (fig. 183). From the bottom of the vertical pipe two or more hollow horizontal arms

extended, at the ends of which were orifices from which the water was discharged. The reaction of the jets caused

the rotation of the machine.

Let H be the available fall measured from the level of the water in the vertical pipe to the centres cf the orifices, r the radius from the axis of rotation to the centres of the orifices, 1.1 the velocity of discharge through the jets, a. the angular velocity of /¢

mummmliml

ll Ii 5 .

1 3

K I lilldi E

mt, f °

- , ', ,; v, ,.~= /» ' :, ;»»

ef if

§ E", :;', ? . ' - l#§ **§ ¢°€ If FIG. 183. the machine. When the machine is at rest the water issues from the orifices with the velocity / (2 gH) (friction being neglected). But when the machine rotates the water in the arms rotates also, and is in the condition of a forced vortex, all the particles having the same angular velocity. Consequently the pressure in the arms at the orifices is H-l-a.2r2/2g ft. of water, and the velocity of discharge through the orifices is 11=~[(2gH-1-a2r2). If the total area of the orifices is w, the quantity discharged from the wheel per second is Q =wv=<.»1/ (2gH-l-azrf).

Uhile the water passes through the orifices with the Velocity 11, the orifices are moving in the opposite direction with the velocity ar. The absolute velocity of the water is therefore 11- ar=/ (2gH+a.'r2)-af.

The momentum generated per second is (GQ/g)(11-ar), which is numerically equal to the force driving the motor at the radius r. The work done by the water in rotating the wheel is therefore (GQ/g) (11-ar)ar foot-pounds per sec.

The work expended by the water fall is GQH foot-pounds per second. Consequently the efficiency of the motor is (11- ar) ar H 2gH -1-aff'-arlar

7' gH T gH

H ZH2

Let ~/2gH+a2f2=af+§ - .

then ' n=1-gH/ 2ar+

which increases towards the limit 1 as ar increases towards infinity. Neglecting friction, therefore, the maximum efficiency is reached when the wheel has an infinitely great velocity of rotation. But this condition is impracticable to realize, and even, at practicable but high velocities of rotation, the friction would considerably reduce the efficiency. Experiment seems to show that the best efficiency is reached when ar =/ (2gH). Then the efhciency apart from friction is 11 = l~/ (2a'1'2)-“flat/gH

=O'4I4G2f2, /gH =o-828,

about 17 % of the energy of the fall being carried away by the water discharged. The actual efficiency realized appears to be about 60 %, so that about 2I % of the energy of the fall is lost in friction, in addition to the energy carried away by the water. § 184. General Stalemenl of Hydrodynamical Principles necessary for the T heory of T urb-ines.

(a) Yhen water flows through any pipe-shaped passage, such as the passage between the vanes of a turbine wheel, the relation between the changes of pressure and velocity is given by Bernoulli's theorem (§ 29). Suppose that, at a section A of such a passage, h1 is the pressure measured in feet of water, 111 the velocity, and Z1 the elevation above any horizontal datum plane, and that at a section B the same quantities are denoted by hz, 112, zz. Then ll-hz = (1122-1112)/2g+Z2*Z1. (I)

If the flow is horizontal, z2=z1; and

h;-hg = (1122-4212)/2g. (Ill)

(b) When there is an abrupt change of section of the passage, or an abrupt change of section of the stream due to a contraction, then, in applying Bernoulli's equation allowance must be made for the loss of head in shock (§ 36). Let 111, 112 be the velocities before and after the abrupt change, then a stream of velocity 111 impinges on a stream at a velocity 112, and the relative velocity is 111-112. The head lost is (111-11»1)2/2g. Then equation (Ia) becomes hz-hi = (U1L'l'12) /2g-(1'1»112)2, /2g:W (2/'1-112) /Z~ (2) To diminish as much as possible the loss of energy from irregular eddying motions, the change of section in the turbine passages must be very gradual, and the curva-A1

A 'wi ture without discontinuity.

3 g (ic)C§ qual1tyfofA/lngalar ggnpulse an range 0 ngu ar amen!

1' 2 tum.-Sup ose that a couple, the

gli I/ { moment of)which1 is M, acts on a 5 //Pl lv body of weight W for t seconds, 1, f 1 during, which it moves from A1

3, I A2 tolA2 (figg $3.96 dLet Ukbé the I /, /' ve ocity o the o y at 1, 112 its Q.-'flb velocity at Az, and let p1, pg be C 'i;~~- a' the perperidiculars from C on 111 3 and 7.11. Then Ml is termed the

w agigular impulse of the couple, and

f' 3 the quantity

" (W/g>(v1l>1~v1l>1l

Vg is the change of angular momen-FIG

184 tum relatively to C. Then, from

the equality of angular impulse

and change of angular momentum

NH = <'U2P2"01P1l 1

or, if the change of momentum is estimated for one second, M = gl (Y'2P1“U1P1>ULICS

Let r1, rg be the radii drawn from C to A1, A2, and let 101, 7.02 be the 112, perpendicular to these radii, making angles Then

111=w1 sec;3;111=1112 sec a;

p1=r1 cos 13; p2=r2 cos a.

= <'w2I'2'”ZU1}'1),

where the moment of the couple is expressed in terms of the radii drawn to the positions of the body at the beginning and end of a second, and the tangential components of its velocity at those oints.

p Now the water fiowing through a turbine enters at the admission surface and leaves at the discharge surface of the wheel, with its angular momentum relatively to the axis of the wheel changed. It therefore exerts a couple -M tending to rotate the wheel, equal and opposite to the couple M which the wheel exerts on the water. Let Q cub. ft. enter and leave the wheel per second, and let *w1, we be the tangential components of the velocity of the water at the receiving and discharging surfaces of the wheel, r1, rg the radii of those surfaces By the principle above,

~ M = (GQ/2) (wm-wm)~ (4)

If a is the angular velocity of the wheel, the work done by the water on the wheel is

T = Ma= (GQ/g) ('w1r1-w2r2)a foot-pounds per second. (5) § 185. Total and Available Fall.-Let H1 be the total difference of level from the head-water to the tail-water surface. Of this total head a portion is expended in overcoming the resistances of the head race, tail race, supply pipe, or other channel conveying the water. Let Y), be that loss of head, which varies with the local conditions in which the turbine is placed. Then

components of 111,

/3 and a with 111, 111.

H = H1-bp

is the available head for working the turbine, and on this the calculations for the turbine should be based. In some cases it is necessary to place the turbine above the tail-water level, and there is then a fall D from the centre of the outlet surface of the turbine to the tail water level which is wasted, but which is properly one of the losses belonging to the turbine itself. In that case the velocities of the water in the turbine should be calculated for a head H-lj, but the efficiency of the turbine for the head H. § 186. Gross Ejicierzcy and Hydraulic Ejiciency of a Turbine.-Let Ta be the useful work done by the turbine, in foot-pounds per second, T1 the work expended in friction of the turbine shaft, gearing, &c., a quantity which varies with the local. conditions in which the turbine is placed. Then the effective work done by the water in the turbine is

T=T¢+T1.

The gross efficiency of the whole arrangement of turbine, races, and transmissive machinery is

171 =T.1/GQH1. (6)

And the hydraulic efficiency of the turbine alone is 11 = T/ GQH ~ (7)

It is this last efficiency only with which the theory of turbines is concerned.

From equations (5) and (7) we get

vGQH = (GQ/g)(w1r1-wmla;

T] = (70171-ZUZV2) G/gH.

This is the fundamental equation in the theory of turbines. In general# 1111 and 1111, the tangential components of the water's motion on entering and leaving the wheel, are completely independent. That the efficiency may be as great as possible, it is obviously necessary that w2=0. In that case 11 = wma/gH. (9)

ar1 is the circumferential velocity of the wheel at the inlet surface. Calling this V1, the equation becomes

71 = w1V1/gH. (ga)

This remarkably simple equation is the fundamental equation in the theory of turbines. It was first given by Reiche (Turbinenbaues, 1877).

§ 187. General Description of a Reaction Turbine.-Professor James Thomson's inward flow or vortex turbine has been selected as the type of reaction turbines. It is one of the best in normal conditions of working, and the mode of regulation introduced is decidedly superior to that in most reaction turbines; Figs. 185 and 186 are external views of the turbine case; figs. 187 and 188 are the corresponding sections;' fig. 189 is the turbine wheel. The example chosen for illustration has suction pipes, which permit the turbine to be placed above the tail-water level. The water enters the turbine by cast-iron supply pipes at A, and is discharged through two suction pipes S, S. The water 1 In general, because when the water leaves the turbine wheel it ceases to act on the machine. If deflecting vanes or a whirlpool are added to a turbine at the discharging side, then 111 may in part depend on 112, and the statement above is no longer true. on entering the case distributes itself through a rectangular ' in equal proportions from each guide-blade passage. It consists supply chamber SC, from which it linds its way equally to the of a centre plate p (ig. 189) keyed on the shaft aa, which passes four guide»blade passages G, G, G, G. In these passages it through stufhng boxes on the suction pipes. On each side ot

acquires a. velocity about equal to tnat due to half the fall, and is ' the centre plate are the curved wheel vanes, on which th- pressure directed into the wheel at an angle of about 10° or 12° with the of the water acts, and the Vanes are bounded on each side by tangent to its circumference. The wheel W receives the water l dished or conical cover plates c, c. Joint-rings j, j on the cover plates make a sufficiently water-tight joint with the casing, to prevent leakage from the guide-blade chamber into the suction pipes. The pressure near the joint rings is not very great, probably not one-fourth the total head. The wheel vanes receive the water without shock, and deliver it into central spaces, from which it flows on either side to the suction pipes. The mode of regulating the power of the turbine is very simple. The guide blades are pivoted to the case at their inner ends, and they are connected by a linkwork, so that they all open and close simultaneously and equally. In this way the area of opening ° through the guide blades is altered without materially altering the angle or the other conditions of enough to adjust the guide-blades very exactly. These turbines are made by Messrs Gilkes & Co. of Kendal Fig. 190 shows another arrangement of a similar turbine, with some adjuncts not shown in the other drawings. In this ease the turbine rotates horizontally, and the turbine case is placed entirely below the tail water. The water is supplied to the turbine by a vertical pipe, over which is a wooden pen trough, containing a strainer, which prevents sticks and other solid bodies getting into the turbine. The turbine rests on three foundation stones, and, the pivot for the vertical shaft being under water, there is a screw and lever arrangement for adjusting it as it wears. The vertical shaft gives motion to the machinery driven by a pair of bevel wheels. On the righ-t are the worm and wheel for working the guide-blade gear. § 188. Hydraulic Power at Niagara.—The largest development of hydraulic power is that at Niagara. The Niagara Falls Power Company have constructed two. power houses on the United States side, the first with IO turbines of 5000 h.p. each, and the second with 10 turbines of 5500 h.p. The effective fall is 136 to 140 ft. In the first power house the turbines are twin outward How reaction turbines with vertical shafts running at 250 revs. per minuteiand ¥ driving the dynamos direct. In the second power house the turbines 5 ' , ,, ,.., , 2 él I m il 17

/ I

i 5

5 K


§ 5

E w Q- (U U7

U I/3

Q- cn F9 D rn ra L*-> U7 ra S-> 5 Q- n O U J cn o F? ru D- P* o cm fn F9 Cl" rv "1 U "4 Er rn >'1-'T D W fi

a 1 ""4= 9 f¥ 17? ¢|

41 ”» Liv? i/'i mei! WL “Z Ati gas |||IlIIlIl|l|| the delivery into the wheel. The guide blade gear may be variously arranged. In this example four linked to the guide/f 0. J C I It ' -llll 'a Y " i 1¢%/ W//fm 5 C W I % FIG. 189. spindles, passing through the case, are Baémahe Piolorl, /I, ,, ,.., z

My/" // »."»/ 1 ' . M, f /J Q

' 1 /
 /
/I
zé Z<<<


/ r

§ ...f “Q am - QQ” ' '&& //f' ww I

//W I W - / . - ' sr-* s', f

g, gg

s Q/ s° gs.-

  • "t

FIG. 191. are inward flow turbines with draft tubes or suction pipes. Fig. 191 shows ~a section of one of these turbines. There is a balancing piston keyed on the shaft, to the under side of which the pressure due to the fall is admitted, so that the weight of turbine, vertical shaft and part of the dynamo is water borne. About 70,000 h.p. is daily distributed electrically from these two power houses. The Canadian Niagara Power Company are erecting a power house to contain eleven units of 10,250 h.p. each, the turbines being twin inward flow reaction turbines. The Electrical Development Company of Ontario are erecting a power house to contain II units of 12,500 h.p. each. The Ontario Power Company are carrying out another scheme for developing 200,000 h.p. by twin inward flow turbines of 12,000 h.p. each. Lastly the Niagara Falls Power and Manufacturing Company on the United States side have a station giving 5,000 h.p. and are constructing another to furnish 100,000 h.p. The mean flow of the Niagara river is about 222,000 cub. ft. per second with a fall of 160 ft. The works in progress if completed will utilize 650,000 h.p. and require 48,000 cub. ft. per second or 21% % of the mean How of the river (Unwin, “ The Niagara Falls Power Stations, ” Proc. Inst. Mech. Eng., 1906). § 189. Dijferent Forms of Turbine Wheel.-The wheel of a turbine f FIG. 190. or part of the machine on which the water acts is an annular space, furnished with curved vanes dividing it into passages exactly or lr iv I OH fhe Outside of the Case' A Worm }vh@<>1 0H one of the, roughly rectangular in cross section. For radial flow turbines the spindles is rotated by a worm d, the motion being thus slow, wheel may have the form A or B, fig. 192, A being most usual with "" A 4 W. I -'r ' 4- — —7, ,~-~- B:gn-...Pg —.

' I I

1 }<~- 7'¢— i - IT - C d <—aw »>-»- 3as A - = = - s ~ ' 'C . ' ' 'T r ~- — 1 1 lin 2 . Ea- . . 5 . . .Q FIG. 192. inward, and B with outward flow turbines. In A the' wheel vanes are fixed on each side of a centre plate keyed on the turbine shaft. The vanes are limited by slightly-coned annular cover plates. In B the vanes are fixed on one side of a disk, keyed on the shaft, and limited by a cover plate parallel to the disk. Parallel flow or axial flow turbines have the wheel as in C. The vanes are limited by two concentric cylinders. Theory of Reaction Turbines. § 190. Velocity of Whirl and Velocity of Flow.-Let acb (fig. 193) be the path of the particles of water in a turbine wheel. That path will be in a plane normal to the f axis of rotation in c radial flow turbines, and on a cylindrical n, surface in axial flow turbines. At any lb point c of the path

I the water will have

" some velocity 11, in

' ' the direction of a

- tangent to the path. v' That velocity may be resolved into two components, a whirling velocity w in the direction of the wheel's rotation at the point c, and a component u at right angles to this, radial in radial flow, and parallel to the axis in axial ow turbines. This second component is termed the velocity of flow. Let v., , wo, u., be the velocity of the water, the whirling velocity and velocity of flow at the outlet surface of the wheel, and vt, wg, vt the same quantities at the inlet surface of the wheel. Let a and B be the angles which the water's direction of motion makes with the direction of motion of the wheel at those surfaces. Then /Q, 1 I Fio. 193. w., =v, , cos;8; u., =v., sin B (lo) wi=v;coso.: ui=v; sin a The velocities of flow are easily ascertained independently from the dimensions of the wheel. The velocities of flow at the inlet and outlet surfaces of the wheel are normal to those surfaces. Let $20, Q; be the areas of the outlet and inlet surfaces of the wheel, and Q the volume of water passing through the wheel per second; then vo =Q/Sb; vi =Q/91. (I I) Using the notation in fig. 191, we have, for an inward flow turbine (neglecting the space occupied by the vanes), Q., =21l'fpdo§ fl; =21l'fidi~ (120) Similarly, for an outward flow turbine, SZ., =21rr., d; SZ; ==2-zrrgd; (1211) and, for an axial flow turbine, Q.. = Qt = 1¥'(7'z2-7't2l- (120) Relative and Common Velocity of the Water and Wheel.-There is another way of resolving the velocity of th€ Waféf- LU be U10 Then the velocity of the water maype resolved into a component V, which the water has V, » in common with the C wheel, and a component of, which is the velocity of the water relatively If (Xw to the wheel. velocity of the wheel at the point c, fig. 194.

I . /I .' ./ » 1, 1 f Velocity of Flow.-It is obvious that the frictional losses of head in the wheel passages will increase as the velocity of flow is greater, that is, the smaller the wheel is made. But if the wheel works under water, the skin friction of the wheel cover increases as the diameter of the wheel is made greater, and in any case the weight of the wheel and consequently the journal friction increase as the wheel is made larger. It is therefore desirable to choose, for the velocity of How, as large a value as is consistent with the condition that the frictional losses in the wheel passages are a small fraction of the total head. The values most commonly assumed in practice are these:- In axial flow turbines, up = 14i=0' 15 to 0'2/ (2gH); In outward flow turbines, ui =o-25/2glH- E), u, ,=o-21 to 0'I7/ 2g(H-bl; In inward flow turbines, u0=u5=0~125/ lzgl-I). § 191. Speed of the Wheel.-The best speed of the wheel depends partly on the frictional losses, which the ordinary theory of turbines VP FIG. 194. I disregards. It is best, therefore, to assume for V., and V; values which experiment has shown to be most advantageous. In axial flow turbines, the clrcumfercntial velocities at the mean radius of the wheel may be taken V, ,=V, =o-6/-Q-I to o-66/°§§ 'll In a radial outward flow turbine, V; =o-56/ 2g(H - 9) Vo = Vin/fi, where r., rf are the radii of the outlet and inlet surfaces. In a radial inward flow turbine, V4 =o-66~/ 2g1-I, Vo=Vxf»/V+ If the wheel were stationary and the water flowed through it, the water would follow paths parallel to the wheel vane curves, at least when the vanes were so close that irregular motion was prevented. Similarly, when the wheel is in motion, the water follows paths relatively to the wheel, which are curves parallel to the wheel vanes. Hence the relative component, v., of the water's motion at c is tangential to a wheel vane curve drawn through the point c. Let ve, V, , 11, -, , be the velocity of the water and its common and relative components at the outlet surface of the wheel, and oi, Vt, vfi be the same quantities at the inlet surface; and let 0 and 4: be the angles the wheel vanes make with the inlet and outlet surfaces' then 7702 = N/ (vroz 'l“Vn2 ' 2VoUro COS Us =/ (1/fs”-l-Va?-2Vwf; cos 0) ' (13) equations which may be used to determine eb and 0. § 192. Condition determining the Angle of the Vanes at the Outlet Surface of the Wheel.-It has been shown that, when the water leaves the wheel, it should have no tangential velocity, if the eflici- f ency is to be as great as possible; ' / that is, 'w., =0. Hence, , from (IO), cos B=O, , B=90°, u., =v.», and if the direction of the water's motion is normal to the outlet surface of the wheel, radial in radial flow, and axial in axial flow turbines. Drawing vs or u., radial or axial as the case may be, and V, - tangential to the direction of motion, uf., can be found by the parallelogram of velocities. From fig. 195, tan ¢='v»/V»=u»/Vi; (14) but ¢> is the angle which the wheel vane makes with the outlet surface of the wheel, which is thus determined when thf velocity of flow un and velocity of the Wheel V, are known. When ¢ is thus determined, fr I, I 'Ya I I 1 I /, W I I / / U'7'o FIG. 195. v., = u, cosec 4: =V., { (1 -I-14,2/VJ). (14a) Correction of the Angle 41 to allow for Thickness of Vanes.-In determining ¢, it is most convenient to calculate its value approximately at first, from a value of 14, obtained by neglecting the thickness of the vanes. As, however, this angle is the most important angle in the turbine, the value should be afterwards corrected to alllow for the vane thickness. et <t>' = t=1r1”'(“»/Vo) = tan~'(Q/SLV0) be the first or approximate value of 4>, and let t be the thickness, and n the number of wheel vanes which reach the outlet surface of the wheel. As the vanes cut the outlet surface approximately at the angle ¢', their width measured on that surface is t cosec ¢'. Hence the space occupied by the vanes on the outlet surface is For A, fig. 192, ntd.. cosec 4> B, fig. 192, ntd cosec 4> ' (15) C, fig. 192, nt(r2-ri) cosec 4> Call this area occupied by the vanes o. Then the true value of the clear discharging outlet of the wheel is SZ.-ov, and the true value of it., is Q/(90 - w), The corrected value of the angle of the vanes will be ¢=tan [Q/Vaffla-w)]. (16) § 193. Head producing Velocity with which the Water enters the Wheel:-Consider the variation of pressure in a wheel passage, which satisfies the condition that the sections change so gradually that there is no loss of head in shock. When the flow is in a horizontal plane, there is no work done by gravity on the water passin through the wheel. In the case of an axial flow turbine, in which the flow is vertical, the fall d between the inlet and outlet surfaces should be taken into account. Let V., V, be the velocities of the wheel at the inlet and l This angle can, if necessary, be corrected to allow for the thickness outlet surfaces,

vt, v., the velocities of the water, ut, ua the velocities of How,

vf¢, v, ,, the relative velocities, ht, h., the pressures, measured in feet of water, rt, r., the radii of the wheel,

at the angular velocity of the wheel. At any point in the path of a portion of water, at radius r, the velocity 21 of the water may be resolved into a component V=a.r equal to the velocity at that point of the wheel, and a relative component ur. Hence the motion of the water maybe considered to consist of two parts:-(a) a motion identical with that in a forced vortex of constant angular velocity a.; (b) a flow along curves parallel to the wheel vane curves. Taking the latter first, 'and using Bernoulli's theorem, the change of pressure due to How through the wheel passages is given by the equation h't +1'ft”/2g = h'., +v~»'/23;

hli h, o = (vroz "7-lri2)/2g°

The variation of pressure due to rotation in a forced vortex is h". -h"., = (Viz -VJ)/2g.

Consequently the 'whole difference of pressure at the inlet and outlet surfaces of the wheel is

hi -h., =h'¢-!-h”, -h', ,—h".,

= (Viz"'Va2)l/2g+(vro2“Uri2)/2g-Case 1. Axial Flow Turbines.~V¢ =V, ; and the first term on the right, in equation 17, disappears. Adding, however, the work of gravity due to a fall of d ft. in passing through the wheel, ht -ht. = (11.02-mf)/2g-d. (170)

Case 2. Outward Flow Turbines.—The inlet radius is less than the outlet radius, and (Viz -V, ,*)/2g is negative. The centrifugal head diminishes the pressure at the inlet surface, and increases the velocity with which the water enters the wheel. Thisvsomewhat increases the frictional loss of head. Further, if the wheel varies in velocity from variations in the useful work done, the quantity (VE-V., ')/2g increases when the turbine speed increases, and vice versa. Consequently the flow into the tur ine increases when the speed increases, and diminishes when the speed diminishes, and this again augments the variation of speed. The action of the centrifugal head in an outward flow turbine is therefore prejudicial to steadiness of motion. For this reason r, ,:r; is made small, generall about 5:4. Even then a governor is sometimes' required to regulate the speed cf the turbine.

Case 3. Inward Flow Turbines.-The inlet radius is greater than the outlet radius, and the centrifugal head diminishes the velocity of flow into the turbine. This tends to diminish the frictional losses, but it has a more important influence in securing steadiness of motion. Any increase of speed diminishes the flow into the turbine, and vice versa. Hence the variation of speed is less than the variation of resistance overcome. In the so-called centre vent wheels in America, the ratio fi: r, , is about 5: 4, and then the influence of the centrifugal head is not very important. Professor James Thomson first pointed out the advantage of a much greater difference of radii. By making fi2Tg=21I, the centrifugal head balances about half the head in the supply chamber. Then the velocity through the guide-blades does not exceed the velocity due to half the fall, and the action of the centrifugal head in securing steadiness of speed is considerable. Since the total head producing flow through the turbine is H-b, and of this h, -h., is expended in overcoming the pressure in the wheel, the velocity of flow i11to the wheel is v, =c, /{2g(l-I-b-(VF-V02/2g+(v, ,,2-vraz)/2g)}, (18) where ov may be taken 0-96.

From (1441),

of the guide-blades. .

§ 195. Condition determining the Angle of the Vanes at the Inlet Surface of the Wheel.-The single condition necessary to be satisfied at the inlet surface of

the wheel is that the

water should enter the

wheel without shock. - -~ -'- 7- » s Y This condition is satis- 9.

fied if the direction of -'

relative motion of the

water and wheel is

parallel to the first

element of the wheel

vanes.

Let A (fig. 196) be a

point on the inlet surace

of the wheel, and

let 1/5 represent in

magnitude and direction

the velocity of the water entering the wheel, and V; the velocity of the wheel. Completing the parallelogram, vrt is the direction of relative motion. Hence the angle between 21,4 and Vt is the angle 0 which the vanes should make with the inlet surface of the wheel. § 196. Example of the Method of designing a Turbine. Professor James Thomsorfs Inward Flow Turbine:- Let H =the available fall after deducting loss of head in pipes and channels from the gross fall;

Q=the supply of water in cubic feet per second; and 1; =the efficiency of the turbine. The Work done per second is 'qGQH, and the horse-power of the turbine is h.p. =1;GQH/550. If 1; is taken at o-75, an allowance will be made for the frictional losses in the turbine, the leakage and the friction of the turbine shaft. Then h.p. =0-o85QH. The velocity of How through the turbine (uncorrected for the space occupied by the vanes and guide-blades) may be taken uf =u»=o~125/ 2gH,

in which case about, ,14th of the energy of the fall is carried away by the water discharged.

The areas of the outlet and inlet surface of the wheel are then Ztffado =2'lI'fadt =Q/0' 1251/ (2ZH)-If we take r., , so that the axial velocity of discharge from the central orifices of the wheel is equal to u., , we get To (Q//V H)1

do = 7'9-

lf, to obtain considerable steadying action of the centrifugal head, rt =2r, ,, then di =%d, ,.

Speed of the Wheel.-Let Vi =o-66/ 2gH, or the speed due to half the fall nearly. Then the number of rotations of the turbine per "

I7 '~ A

s —

0| OG-i 1

FIG. 196.

second is

N =Vi/27ffg = I'O579/ (Hxf H/Q);

also Va =Vtr../rt =0~33~/ 2,25-Angle of Vanes with Outlet Surface.

Tan ¢ =ua/V.. =o~ 125/o-33 = -3788; ¢> =2I ° nearly.

If this value is revised for the vane thickness it will ordinarily become about 25°.

Velocity with which the Water enters the Wheel.-The head producing the velocity is,

H - (VF/2g)(1 +1402/VF) +1¢-1”/22

=H{1 — I.t1356(1-l-o-o358)+-o156}

=o~5646

Then the velocity is

U.. =v.~/ <1 +u.2/ve). Yf = ~96~/ 2S<'S646H> =°'721~/ HHlt will be shown immediately that Angle of Guide-Blades v~;=utcosec0; " I Sin-y=u¢/°v¢=o~125/o~721=o-173; or, as thisfis olnly a small term, and 0 is on the average 90°, we 'y=IO° nearly. t - f .= . .

may a e, or the present purpose, if., u, nearly. nserting these values, and remembering that for an axial How turbine Vi =V., , l) =o, and the fall d in the wheel is to be added, l.

For an outward flow turbine, E

s si

For an inward flow turbine,

< an

§ 194. Anfle which the Guide-Blades make .with the Circumference of the Whee .—At the moment the water enters the wheel, the radial component of the velocity is ut, and the velocity is vg. Hence, fh'y is the angle between the guide-blades and a tangent to the w ee

1 =Sir1"(ut/vt)-Tangential

Velocity of Water entering Wheel.

wg évfcos 'y =o~7101/El.

Angle of Vanes at Inlet Surface.

Cot 0 =(wt-Vt)/u;=(~7101 -~66)/-125 = -4008; 0 =68° nearly. »

Hydraulic Ejiciency of Wheel.

-q=wtVa/gH==-7I0IX'66X2

=0'9373-

This, however, neglects the friction of wheel covers and leakage. The efficiency from experiment has been found to be 0-75 to o~8o. Impulse and Partial Admission T urbines. § 197. The principal defect of most turbines with complete admission is the imperfection of the arrangements for working with less than the normal supply. With many forms of reaction turbine the efficiency is considerably reduced when the regulating sluices are partially closed, but it is exactly when the supply of water is deficient that it is most important to get out of it the greatest ossible amount of work. The im erection of P P the regulating arrangements is therefore, from the practical point of view, a serious defect. All turbine makers have sought by various methods to improve the regulating mechanism. B. Fourneyron, by dividing his wheel by horizontal diaphragms, virtually obtained three or more separate radial iiow turbines, which could be successively set in action at their full power, but the arrangement is not altogether successful, because of the spreading of the water in the space between the wheel and guide-blades. Fontaine similarly employed two concentric axial flow turbines formed in the same casing. One was worked at full power, the other regulated. By this arrangement the loss of efficiency due to the action of the regulating sluice affected only half the water power. Many makers have adopted the expedient of erecting two or three separate turbines on the same waterfall. Then one or more could be put out of action and the others worked at full power. All these methods are rather palliatives than remedies. The movable guide-blades of Professor James Thomson meet the difficulty directly, but they are not applicable to every form of turbine. C. Callon, in x840, patented an arrangement of sluices for axial or outward flow turbines, which were to be closed successively as the wa tk supply diminished. By preference the sluices were closed by pairs, two diametrically opposite sluices forming a pair. The water was thus admitted to opposite but equal arcs of the wheel, and the forces driving the turbine were symmetrically placed. As soon as this arrangement was adopted, portion of the sluice, and stopped each time it passes a closed portion of the sluice. It is thus put into motion and stopped twice in each rotation. This gives rise to violent eddying motions and great loss of energy in shock. To prevent this, the turbine wheel with partial admission must 'be placed above the tail water, and the wheel passagesmbe allowed to clear themselves of water, while passing from one open portion of the sluices to the next. . i y But if thewheel passages are free of water when they arrive at the open guide passages, then there can be no pressure other than atmospheric pressure in the clearance space between guides and wheel. The water must issue from the sluices with the whole velocity due to the head; received on the curved vanes of the wheel, the jets must be gradually deviated and discharged with a small final velocity only, precisely in the same Way, as when a single jet strikes a curved vane in the free air. Turbines of this kind' are therefore 'termed turbines of free deviation. There is no variation of pressure in the jet during the whole time of its action On 'the wheel, and the whole energy of the jet is imparted to the wheel, simply by the impulse due to its gradual change of momentum. It is clear that the water may be admitted in exactly the same way to any fraction of the circumference at pleasure, without altering the efhciency of the wheel. The diameter of the wheel may be made as large as convenient, and the water admitted to a small fraction of the circumference only. Then the number of revolutions is independent of the water velocity, and may be kept down to a manageable value. t § 198. General Description of an Impulse Turbine or Turbine with free Deviation.-Fig, 197 shows a general sectional elevation 'of a (Jirard turbine, in 1/ ah', ¢, . gg»r;y?, ; “" ' »é1*""r'" if L; lla tid! is rs. which the flow is axial. The water, admitted above a horizontal ffloor, passes down through the annular wheel containing the guide» blades (3, G, and 14 if v* » a a, Q it I

1 1 l, f rf '; ~ at thence into the revolving wheel. WW. The revolving wheel is fixed to a hollow shaft suspended from the pivot p. The solid interrlal shaft .rs .is merely a fixed column WMM The advantage pf this is that .the ivot is Fic.. 198. accessible for lubrication and adjustment. B is the mortise bevel wheel by which the power of the turbine is given off. The sluices 31, | - . c Ilfw ls l. ., . ., . .

. . l. - - "' -» V" *-'-5° I

r- l;=:e girl g ~-~ r, T' 1 lil na .1 'T — sa 'Q ii fiig, — - “ .., ... 2 2 T El' il!! flies.;-fs > Il *E* iii o 5 ia esse L , ,, , a modification of the mode of action of the water in the turbine became necessary. If the turbine wheel passages remain full of y water during the whole rotation, the water contained in each passage must be put into motion each time it passes an open are worked by the hand wheel h, which raises them successively, in a way to be described presently. d, d are the. sluice rods. Figs. 198, 199 show the sectional form of the guide-blade chamber and wheel and the curves of the wheel vanes and guide-blades, when drawn on a plane development of the cylindrical section of the Q Wheel; a, ct, a are the sluices for cutting off the water; b, b, b are, . apertures by which the E. -~—~, %, ,.|i. . entrance or exit of air is facilitated as the Q buckets empty and fill. Figs. zoo, 201 show the guide-blade gear. a, a, a p are the sluice rods as before. At the top of each sluice rod is a 6 'n small block c, having a project in ton uc, ,, - which slidei in gthe groove of the circular Q/ cam plate d, d. This ' ' Y '1 '-L-"f " -f"circular plate is sup- FIG 199 ported on the frame e, and revolves on it bymeans of the flanged rollers f. Inside, at the top, the cam plate is toothed, and gears into a spur pinion connected with the hand wheel h. At gg is an inclined groove or shunt. When the tongues of the blocks c, c arrive at g, they slide up to a second groove, or the reverse, accordin as the cam plate is revolved in one direction or in the other. As giis operation takes place with each sluice successively, any number of sluices can be opened or closed as desired. The turbine is of 48 horse power on 5~12 ft. fall, and the supply of water varies from 35 to 112 cub. ft. per second. The Hence the vane angle at inlet surface is given by the equation cot0 = (wi -V.)/ug = (0-82-0-5)/o~45 = ~71; 0=55°. The relative velocity of the water striking the vane at the inlet

FIG. 200. efficiency in normal working is given as 73 %. The mean diameter of the wheel is 6 ft., and the speed 27-4 revolutions per minute. As an example of a partial admission radial How impulse turbine, a 100 h.p. turbine at Immenstadt may be taken. The fall varies from 538 to 570 ft. The external diameter of the wheel is 4% ft., and

F1o. 201. its internal diameter 3 ft. IO in. Normal speed 400 revs. per minute. Water is discharged into the wheel by a single nozzle, shown in fig. 202 with its regulating apparatus and some of the vanes. The water enters the wheel Y l e enf=;i§ 'ea€:.ii W % A tion of motion, & '® ' Zggle oilithe viiiireiil N § vanesis2o°. The

 § efficiency on trial
' &~ 199. Theory

of ythe Impulse I, ""' Turbine.-The / / theory of the im/ / g ' pulse turbine Q Z i g does not essen-Q Q tially differ from 2 Q, % that of 'the re° —' action turbine, except that there is no ressure in FIG- 202- the wlgeel opposing the discharge from the guide-blades. Hence the velocity with enters the wheel is simply U4 =0°96/ 2.§ (H ~ 5), which the water where l) is the height of the top of the wheel above the tail water. If the hydro pneumatic system is used, then b=0. Let Q". be the maximum supply of water, rl, rg the internal and external radii of the wheel at the inlet surface; then ui = Q, "/i1|'(f22-712)l. The value of u; may be about 0'45V 2g(Fl -5), whence rl, rg can be determined. The guide-blade angle is then given by the equation . Sin v =u</ve =0-45/0°94= -48; v = 29°- The value of ut should, however, be corrected for the space occupied by the guide-blades. The tangential velocity of the entering water is ru; =v; cos -y =o-82»/ zgl H -li i. The circumferential velocity of the wheel may be (at mean radius) Ve =o-5~/ 2gZH-5). edge is 14, =u¢ cosec0== 1-22u;. This relative velocity remains unchanged during the passage of the water over the vane; consequently the relative velocity at the point of discharge is vm = 1-22u;. Also in an axial flow turbine V., =V¢. If the final velocity of the water is axial, then cos¢ =V0/v, -, ,=V, /vfi =0°5/(I'22 X0-45) =cos 24° 23'. This should be corrected for the vane thickness. Neglectmg this, ua =v, ., sin 4: =v, ; sin 4> =u; cosec 0 sin ¢ =o-5u;. The discharging area of the wheel must therefore be greater than the inlet area in the ratio of at least 2 to 1. In some actual turbines the ratio is 7 to 3. This greater outlet area is obtained by splaying the wheel, as shown in the section (fig. 199). § 200. Pelton Wheel.-In the mining district of California about 186O simple impulse wheels were used, termed hurdy-gurdy wheels. The wheels rotated in a vertical plane, being supported on a horizontal axis. Round the circumference were fixed flat vanes which were struck normall by a jet from a nozzle of size varying with the head and quantity of, water. Such wheels have in fact long been used. They are not efficient, but they are very simply constructed. Then attempts were made to improve the efficiency, first by using hemispherical cup vanes, and] then by using a double cup vane with a central dividing ridge, an arrangement invented by Pelton. In this last form the water from the nozzle passes half to each side of the wheel, just escaping clear of the backs of the advancing buckets. F ig. 203 shows a Pelton vane. Some small modifications have been made by other makers, but they are not of any great importance. Fig. 204 shows a complete Pelton wheel with frame and casing, supply pipe and nozzle. Pelton wheels have been very largely used in America and to some extent in Europe. They are extremely simple and easy to construct or repair and on falls of 100 ft. or more are very efficient. The jet strikes taiientially to the mean radius of the buckets, and the face of the buckets is not quite radial but at right angles to the direction of the jet at the point of first impact. For greatest efficiency the peripheral velocity of the wheel at the mean radius of the buckets should be a little less than half the velocity of the jet. As the radius of the wheel can be taken arbitrarily, the number of revolutions per minute can be accommodated to that of the machinery to be driven. Pelton wheels have been made as small

-;

/f -, E?,2' .- - ' . . 1 if ~x § o FIG. 203. i;'-ip W ie ae » /2 ' W /./O - 12 7 ~ gt qs. ng e ¢' <1/M - » ==V' ': -in f f. s§ 14!» v g|; -~§ ii: r “ » FIG.'f2o4. V as 4 in. diameter, for driving sewing machines, and as large as 24 ft. The efficiency on high falls is about 80 %.- When large power is required two or three nozzles are used delivering on one wheel. The width of the buckets should be not less than seven times the diameter of the jet. At the Comstock mines, Nevada, there is a 36-in. Pelton wheel made of a solid steel disk with phosphor bronze buckets riveted to the rim. The head is 2100 ft. and the wheel makes 1150 revolutions per minute, the peripheral velocity being 180 ft. per sec. With a é-in. nozzle the wheel uses 32 cub. ft. of water per minute and develops loo h.p. At the Chollarshaft, Nevada, there are six Pelton wheels on a fall of 1680 ft. driving electrical generators. With § -in. nozzles each develops 125 h.p. § 201. Theory of the Pelton Wheel.-Suppose a jet with a velocity v strikes tangentially a curved vane AB (fig. 205) moving in the same direction with the velocity u. The water will flow over the vane with the relative velocity v-u and at B will have the tangential relative velocity v-u making an angle gr with the direction of the vane's motion. Combining this with the velocity u of the vane, the absolute velocity of the water leaving the vane will bew § Bc. The component of -w in the direction of motion of the vane is Ba=Bb4ab =u-(11-u) cos ~a.' Hence

if Q is the quantity of

water reaching 'the vane

per second the changehof momentum per second in

the direction of the vanc's B a 6 motion is (GS/g)[v-{uv-u)¢0S°1}l=(/g)(v-u) cos a). R a.=0°,

cos a.=I, and the 'change

of momentum per second,

which is eq-ual to the

effort driving the vane, is

/4* P=2(GQ/g)(z/-u). The

work done on the vane is

Pu=2(GQ/g)(v-u)u. If a

series of vanes are interposed

in succession, the

quantity of water impinging

on the vanes per second is the total discharge of the' nozzle, and the energy expended at the nozzle is GQv”/zg. Hence the efficiency of the arrangement is, when a =0°, neglecting friction, 1/=2P1¢/GQv'=4(v-"lu/v'.

which is a maximum and equal to unity if u=%v. In that case the whole energy of the jet is usefully expended in driving the series of vanes. In practice a cannot be quite zero or the water leaving one vane would strike the back of the negt advancing vane. Fig. 203 shows a Pelton vane. The water divides each way, and leaves the vane on each side in a direction nearly parallel to the direction of motion of the vane. The best velocity of the vane is very approximately half the velocity of the jet.

§ 202. Regulation of the Pelton Wheel.-At first Pelton wheels were adjusted to varying loads merely by throttling the supply. This method involves a total loss of part of the head at the sluice or throttle valve. In addition as the working head is reduced, the relation between wheel velocity and jet velocity is no longer that of greatest efficiency. Next a plan was adopted of deflecting the jet so that only part of the water reached the wheel when the load was reduced, the rest going to waste. This involved the use of an equal quantity of water for large and small loads, butit had, what in some cases is an advantage, the effect of preventing any water hammer in the supply pipe due to the action of the regulator. In most cases now regulation is effected by varying the section of the jet. A conical needle in the nozzle can be advanced or withdrawn so as to occupy more or less of the aperture of the nozzle. Such a needle can be controlled by an ordinary governor. U 4/

""';£:f?

H

FIG. 205.

§ 2o3. General Considerations on the Choice of a Type of Turbine.-The circumferential speed of any turbine is necessarily a fraction of the initial velocity of the water, and therefore is greater as the head is greater. In reaction turbines with complete admission ~the number of revolutions per minute becomes inconveniently great, for the diameter cannot be increased beyond certain limits without greatly reducing the efficiency. In impulse turbines with partial admission the diameter can be chosen arbitrarily and the number of revolutions kept down. on high falls to any desired amount. Hence broadly reaction turbines are better and less costly on low falls, and impulse turbines on high falls. For variable water How impulse; turbines have some advantage, being more efficiently regulated. On the other hand, impulse turbines lose etiiciency seriously if their speed varies from the normal speed due to the head. If the head is very variable, as it often is on low falls, and the turbine must run at the same speed whatever the head, the impulse turbine is not suitable. Reaction turbines can be constructed so as to overcome this difficulty to a great extent. Axial How turbines with vertical shafts have the disadvantage that in addition to the weight of the turbine there is an unbalanced water pressure to be carried by the footstep or collar bearing. In radial flow turbines the hydraulic pressures are balanced. The application of turbines to drive dynamos directly has involved some new conditions. The electrical engineer generally desires a high speed of rotation, and a very constant speed at all times. The reaction turbine is generally more suitable than the impulse turbine. As the diameter of the turbine depends on the quantity of water and cannot be much varied without great.ineEiciency, a. difficulty arises on low falls. This has been, met by constructing four independent reaction turbines on the same shaft, each having of course the di-ameter suitable for one-quarter of the whole discharge, and having a higher speed of rotation than a larger turbine. The turbines at Rheinfelden and Chevres are so constructed. » To ensure constant speed of rotation when the head varies considerably without serious inefficiency, an axial flow turbine is generally used. It is constructed of three or four concentric rings of vanes, with independent regulating sluices, forming practically independent turbines of different radii. Any one of these or any combination can be used* according to the state of the water. With a high fall the turbine of largest radius only is used, and the speed of rotation is less than with a turbine of smaller radius. On the other hand, as the fall decreases the inner turbines are used either singly or together, according to the power required. .At the Ztirich Waterworks there are turbines of go h.p. on a fall varying from rcs ft. to 4% ft. The power and speed are kept constant. Each turbine has three concentric rings. The outermost ring gives go h.p. with 105 cub. it. per second and the maximum fall. The outer and middle compartments 'give the same power with 140 cub. ft. per second and a fall of 7 ft. ro in. All three compartments working together develop the power with about 250 cub. ft. per second. In some tests the efficiency was 74% with the outer ring working alone, 7 5.4% with the outer and middle ring working and a. fall of 7 ft., and 80-7 % with all the rings working. V I § 204. Speed Go-uerning.»-When turbines are used to drive dynamos direct, the question of speed regulation is of great importance. Steam engines using a light elastic fluid can be easily regulated by governors acting on throttle or expansion valves. It is different with water turbines using a fluid of great inertia.. 2

U i I

Q

L .. f .

li A

~, "" 1351 . .... w

-~- {@» M( f., ,, -0,1

vs

7 R ..., . . . ., .. ., , ., .

I ° .-»I ..

=, -

E3.

nl cy

W .

~ .

N.
- '|'

ty j '&

/ i"

(EJ hai

I || *

m @! | .

4 'iiill — r R"””"

Flo. 206.

In one of the Niagara pen stocks there are 400 tons of water flowing at ro ft. per second, opposing enormous resistance to rapid h change of speed of How. The sluices of water turbines also are necessarily large and heavy. Hence relay governors must be used, and the .tendency of relay governors to hunt .must be overcome. In the Niaga.ra'Falls Power House No. x, each turbine has a very sensitive centrifugal governor acting on a ratchet relay. The governor puts into gear one or other of-two ratchets driven by the turbine itself. According as one or the other ratchet is in gear the sluices are raised or lowered. By a subsidiary arrangement the ratchets are gradually put out of gear unless the governor puts them in gear again, and this prevents- the over correction of the speed from the lag in. the action of the governor. In the Niagara Power House -No. 2, the relay 'is an hydraulic relay similar in principle, but rather more complicated in arrangement, to that shown in figi 2o6, which is a' governor used for the 12 50 h.p. turbines at Lyons. The sensitive governor G opens a valve and puts into action a plunger driven 'by oil pressure from an oil reservoir. As the plunger moves forward it gradually closes the oil admission valve by lowering the fulcrum end f of the valve lever which rests on awedge watt ached to the plunger. If the speed is still too high, the governor reopens the valve. In the case of the Niagaraturbines the oil pressure is rzoo lb per sq. in. One millimetre of movement of the governor sleeve completely opens the relay valve, and the relay plunger exerts a force of 5o tons. Thesluices can be completely opened or shut in twelve seconds. The ordinary variation of speed of the turbine with' varying load does not exceed 1 %. If all the load is thrown off, the momentary variation of speed is not more than 5%. To prevent hydraulic shock in the supply pipes, a relief valve is provided which opens if the pressure is in excess of that due to theehead. - "5 . § 205. The Hydraulic Ram.-The hydraulic ram is an arrangement by which a quantity of water falling a distance 'h forces a portion of the water to rise to a height hr, greater than h. It consists of a supply reservoir (A, fig. 207), into which the water enters from some natural stream. A pipe s of considerable length conducts the water to a lower level, where it is discharged intermittently through a self-acting pulsating valveyat d. The supply pipe s may be htted with a flap valve for stopping the ram, and this is attached in some .cases to afloat, so that the ram starts and stops itself automatically; according the supply cistern fills or empties. The lower fioatis just“suH'leient=.to keep open the Hap after it has been raised the action of the upper float. The length of chain is adjusted so that the upper float opens the flap when the level in the cistern is at the desired height. If the water-level falls below the lower iloat the Hap closes. The pipe .r should be as long and as straight as possible, and as it is subjected to considerable pressure from the sudden arrest of the motion of the water, it must be strong and strongly

4 PUMPS this opens an aperture fin. in'diameter, made in a brass screw plug b., The hole is reduced to 115 in. in diameter at the outer end of the plug and is closedby a small valve opening inwards. Through this, during the rebound after each stroke of the ram, a small quantity of air is sucked in which keeps the air vessel supplied with its elastic cushion of air. During the recoil aiteria sudden closing of the valve d, the pressure below it is diminished and the valve opens, permitting outflow. In consequence of the .flow through this valve, the water in the supply pipe acquires a gradually increasing velocity. The upward How of the water, towards the valve d, increases the pressure tending to lift the valve, and at last, 2 if the valve is not too » I ®@&¥s¥sog

V ix

s - S heavy, lifts and closes it. The forward mo» mentum of the column in- the “supply pipe being destroyed by the ' 7: stoppage of the flow, the water exerts a | § s rt S P Mm, f pressure at the end of the pipe sufficient to open the delivery valve o, and to cause ~ <' " * if " -if~'il¥'; li; 5 a portion of the water M to flow into the air vessel. As the water in the supply, pipe comes to"rest and iiecoils, thelvalve rl opens again and the » Part of the energy of the descending

sf” '7, Y s ' » x .- I v /, // // 5 § 1 ~ . / H u v I V K!

2 I t

Yhlh / if '.*.», C #4 / H j/ -. *" //, . w. .... ta ..¢i.t.z....LA', %wA, ,, ;, g, w

I ~ F IG. 208.operation is- repeated. coiumn is employed in compressing the air at the end of the supply pipe and expanding the pipe itself. This causes a recoil of the water which momentarily diminishes the pressure in the pipe below the pressure due to the statical head. ~ This assists in opening the valve d. The recoil of the water is sufficiently great to enable a pump to be attached to the ram body instead of the direct rising pipe. With this arrangement a ram working with muddy water may be employed to raise clear spring water. Instead of lifting the delivery valve as in the ordinary ram, the momentum of the column drives a sliding or elastic piston, and the recoil brings it back. This piston lifts and forces A alternately the clear water through ordinary 737 ze.. A " pump valves' PUMPS T °' - - T. “ ' ! f A"' - .

-'L-E-I.-I~'I-eiei- ~ ' - 4 ' , y § 206. The different classes of pumps correlfii-ff-';-f-;-f-

-' Q ' - spond almost exactly to the different classes -: -::;:'::'; ' f ' of water motors, although the mechanical 4;3-'-'- -'- -'- - -°~'3 ' " 1” 9 ~ =. ' details of the construction are somewhat ~ - - - * - ~ -; - /711.1 » . ¢ 61|-~ d-5 t Th 1 d . ~ a 7/T ywy/7: X 1 |;, Q A 1 eren ey are proper y reverse - water 8 // Flo. 207. jointed. a is an air vessel, and e the delivery leading to the reservoir at a higher level than A, into which water is to be pumped. Fig. 2o8 shows in section the construction of the ram itself. d is the pulsating discharge valve already mentioned, which opens inwards and downwards. The stroke of the valve is regulated by the cotter through the spindle, under which are washers by which the amount of fall can be regulated. At o is a delivery valve, opening outwards, which is often a ball valve but sometimes a flap-Valve. The water which is pumped passes through this valve into the air vessel a, from which it flows by the delivery pipe in a regular stream into the cistern to which the water is to be raised., In the vertical chamber behind the outer valve a small air vessel is formed, and' into f » | i motors. Ordinary reciprocating pumps Corre=. ¥ s

sg . /ll! spond to water-pressure engines. Chain and bucket pumps are in principle similar to water wheels in which the water acts by weight. Scoop wheels are similar to undershot water wheels, and centrifugal pumps to turbines. Reciprocating Pumps are single or double acting, and diiieg from water-pressure engines in that the valves are moved by the 'water instead of by automatic machinery. ' They may be classed thus:-~ ' tr. Lift Pumps.~Thc water drawn through a foot valve on the ascent of the pump bucket is forced through the bucket valve when it descends, and lifted by the bucket when it reascends. Such pumps- give an intermittent discharge. 2. Plemger or Force Pumps, in which the water drawn through the foot valveis displaced by the descent of a solid plunger, and forced through a delivery valve. They have the advantage that the friction is less than that of lift pumps, and the packing round the plunger is easily accessible, whilst that round a lift pump bucket is not. The flow is intermittent. 3. The Double-acting Force Pump is in principle a double plunger pump. The discharge fluctuates from zero to a maximum and back to zero each stroke, but is not arrested for any appreciable time. 4. Bucket and Plunger Pumps consist of a lift pump bucket combined with a plunger of half its area. The flow varies as in a double-acting pump. 5. Diaphragm Pumps have been used, in which the solid plunger is replaced by an elastic diaphragm, alternately depressed into and raised out of a cylinder. were found by Morin to have an efficiency of 16 to 40%, or on the average 2 5%. When used to pump water at considerable pressure, through hose pipes, the efficiency rose to from 28 to 57%, or on the average, with 50 to loo ft. of lift, about 50%. A large pump with barrels 18 in. diameter, at speeds under.6o ft. per minute, gave the following results:—Lift in feet . . 14% ' 34 47 Efficiency ... ~46 -66 -70 The very large steam-pumps employed, for Waterworks, with 150 ft. or more of lift, appear to reach an efficiency of QO(%;, not including the friction of the discharge pipes. Reclconed on the indicated work of the steam-engine' the efficiency may be 80%. - Many small pumps are now driven electrically and are usually three-throw single-acting pumps driven from the electric motor by gearing. It is not convenient to vary the speed of the motor to accommodate it to the varying rate of pumping usually required. Messrs Hayward Tyler have introduced a mechanism for varying the stroke of the pumps (Sinclair's patent) from full stroke to nil, without stopping the pumps. § 207. Centrifugal Pump.-For large volumes of water on lifts not exceeding about 60 ft. the most convenient pump is the centrifugal pump. Recent improvements have made. it available also for very high lifts. It consists of a wheel or fan with curved vanes enclosed in an annular chamber. Water flows in at the centre and is discharged at the periphery. The fan may rotate in a vertical or horizontal plane and the water may enter on one or both sides of the fan. In the latter case there is no axial unbalanced pressure. The fan and its casing 'must be hlled with water before it can start, so that if not drowned there must be a foot valve on the suction pipe. When no special, attention needs to be paid to efficiency the water may have a velocity of 6 to 7 ft. in the suction and delivery pipes. The fan often has 6 to 12 vanes. For a double-inlet. fan of diameter D, the diameter of the inlets is D/2. If, Q is the discharge in cub. ft. per second D=about o~6 /Q in average cases. The mo'-~»-4:ro '0 Q as"*5es 92 <°3'l°, ;;'23”S'=15='§ 9*~E

1

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"cn, :'$s=>o.,5~ gms:-c:'<, =;-g 3§ , . €w%-U.-iff; 3, ,, § '°5 9-°»~»<2£9~E! U, § ;, r39, -5 5 ya Egs?n agsawv eéwawaa Y § =§ ; ~, , ~o ; »§ : § - § "' ' § ve" '§ e* “ %®l§ & > s*, $ S § t § s»<§ :;“ ' ' °::; 11 § 3 § i. ' C; 2 § tw§ § I. I A -;' R W § ~l.r' lie' €~f, ¢% ' ' L 5 5 . ///u , i s., ,/ § twmmW 1'1".' ° ' Q § / , 1/, / Nxs.//M/ &“§ § au » !'$'§ ! Ns § W§ ~&~|&, c FIG. 209. and the disk is keyed on the driving shaft C. The casing A has a spirally enlarging discharge passage into the discharge pipe K. A cover L gives access to the pump. S is the suction pipe which opens into the pump disk on both, sides at D. Fig. 210 shows a centrifugal pump differing from ordinary centrifugal pumps in one feature only. The water rises through a suction pipe S, which divides so as to enter the pump wheel W at the centre on each side. The pump disk or wheel is very similar to a turbine wheel. It is keyed on a shaft driven by a belt on a fast and loose pulley arrangement at P. The water rotating in the pump disk presses outwards, and if the speed is sufficient a continuous flow is maintained through the pump and into the discharge pipe D. The special feature in this pump is that thewater, discharged by the pump disk with a whirling yelocity of not inconsiderable magnitude, is allowed to continue rotation in a chamber somewhat larger than the pump. The use of this whirlpool chamber was first suggested by Professor James Thomson. It utilizes the energy due to the whirling velocity of the water which in most pumps is wasted in eddies in the discharge pipe. In the pump shown guide-blades are also added which have the direction of the stream lines in a free vortex. They dofnot therefore interfere with the action of the water when pumping the normal quantity, but only prevent irregular motion. At A is a plug by which the pump case' is filled before starting. If the pump is above the water, to be pumped, a foot valve is required to permit thepump to be filled. Sometimes instead of the foot valve a delivery valve is' used, an air-pump or steam jet pump being employed to exhaust the air from the pump case. § 208. Design and Proportions of a Centrifugal Pumpf-The design of the pump disk is ver simple. Let n, r., be the radii of the inlet and outlet surfaces of the pump disk, df, dn the clear axial width at those radii. The velocity of flow through the pump may be taken

Fro. 210. same as for a turbine. If Q is the quantity pumped, and H the t, u¢=0'25/2 frf. ' (1) 21rr;d¢ = Also in practice de = I'2fi . . Hence, (2) fi ='257I/ (Q/V H). Usually ro = 271, and d., =d; or § d; . according as the disk is parallel-sided or coned. The water enters the wheel radially with the velocity ur, and ua = Q/21rr, , d., . QS? Fig. 21 I shows the notation adopted for the velocities. Suppose the water enters the wheel with the velocity vg, while

FIG. 211 the velocity of the wheel is Va. Completing »the parallelogram, va is the relative velocity of the water and wheel, and is the proper direction of the wheel vanes. Also, by resolving, us and ws are the component velocities of flow and velocities of whir of the velocity vt bf the water. At the outlet surface, v., is the final; velocity of discharge, and the rest of the notation is similar to that for the inlet surface. Usually the water flows equally in a ll directions in the eye of the wheel, in that case va is radial. Then, in normal conditions of working, at the inlet surface, 11; =1t; wg =0 tan 8=us/Vt (4) v, ¢ = ug cosec 6 = 4 iu¢2 +V, ”, If the ump is raising less or more than its proper quantity, 0 will not satis y the last condition, and there is then some loss of head in F shock. At the outer circumference of the wheel or outlet surface, 11,0 = u., coscc ¢ w, ,=V..-u. cot q> va 5 J iun2'i' (Vu “ua C()t Variation of Pressure in the Pum Disk.+Precisely as in the case 1 Y (5) of turbines, it can be shown that tllie variation of pressure between the inlet and outlet surfaces of the pump is ho"hi = (Vo2*Vi2)!2g (vroz vr'i2) Izg- ' i, Inserting the values of vm. v, ; in (4) and (5), we get for normal conditions of working ll., -ll; = (V, ,'-VF)/2g -11,2 cosec2¢/2g+ (uf -I-VF)/2g =V.,2/2g -uv” coscc 24>/2g-5-11,2/2g. (6) Hydraulic Ejiciency of the Pump.—Neglecting disk friction, lournal friction, and leakage, the efficiency of the pump can be found ln the same way as that of turbines (§ 186). Let M be the moment of the couple 'rotating the ump, and o. its angular velocity; wa, 1, the tangential velocity ofp the water and radius at the outlet surface; wr, rt the same quantities at the inlet surface. Q being the discharge per second, the change of angular momentum per second is (GQ/K) (wir. -wirel-Hence M = (GQ/g) (won, -win). In normal working, wg =o. Also, multiplying by the angular velocity, the work done per second is Ma = (GQ/g)w, , r., a.. But the useful work done in pumping is GQH. Therefore the efficiency is 1| =GQH/Ma. =gH/w, , r, , a =gH/'w0V, ,. (7) § 209. Case I. Centrifugal Pump with no Whirlpool Chamber.-When no special provision is made to utilize the energy of motion of the water leaving the wheel, and the pump discharges directly into a. chamber in which the water is flowing to the discharge pipe, nearly the whole of the energy of the water leaving the disk is wasted. The water leaves the disk with the more or less considerable velocity vo, and impinges on a mass flowing to the discharge pipe at the much slower velocity 11.. The radial component of 'vs is almost necessarily wasted. From the tangential component there is a gain of pressure (w.»='-v?)/2g - (wa-°vS>“/2g =vs(wa°“vs)/grwhich will be small, if v, is small compared with wo. Its greatest value, if -v. =$w¢, , is éwaf/2g, which will always be a small part of the whole head. Suppose this neglected. The whole variation of pressure in the pump disk then balances the lift and the head 14,42/2g necessary to give the initial velocity of flow in the eye of the wheel. =vv2/2g uu2 COSQC 2¢/2g+ui2/2g1 H =V, ,2/2g-u, ,2 coscc 2¢»/2gQ (8) or V0 = 4 (2gH +1402 coscc '4> 5 and the efficiency of the pump is, from (7), n=gH/V..w..=gH/{V(V.»-mcot 4>)}. ', = (V02-us” cosec ”¢>), /{2V., (V0 - up cot ¢}, (9). For ¢=90°, 1z=('¢F-1192)/2/V, . which is necessarily less than é. That is, half the work expended in driving the pump is wasted. By re curving the vanes, a plan introduced by Appold, the efficiency is increased, because the velocity v., of discharge from the pump is diminished. If ¢ is very small, cosec ¢ = cot ¢; and then n = (Vo-4-ua Cosec ¢>)/2V0, T which may approach the value I, as qb tends towards 0. Equation (8) shows that. us cosec 4>-cannot be greater than V, ,. Putting u, , =0'254 (2gH) we get the following numerical values of the efficiency and the circumferential velocity of the pump:ff

90° o-4; 1-gg# 2gl-I 45 0'5 1°-333 3-65 I:I2, , 0 73 1 24 .. I0 0'84 I'75 11 4> cannot practically be made less than 20°; and, allowing for the frictional losses neglected, the efficiency of a pump in which ¢ =20° is found to be about -60. § 210. Case 2. Pump with a Whirlpool Chamber, as in fig. 210.-Professor James Thomson first suggested that the energy of the water after leavin the pump disk might be utilized, if a space were left in which a free vortex could be formed. In such a free vortex the velocity varies inversely as the radius. The gain of pressure in the vortex chamber .is, putting ra, r., for the radii to the outlet surface of wheel and to outside of free vortex, 7102 102- 'ULF 2 2g(I ni) 28 (I ki if k =1../Y... The lift is then, adding this to the lift in the last case, H = {V, ,'-uc' COS€C2¢'f'U¢,2(I -k')}/zg. But of =V., L2V, , u., cot ¢+u.,2 cosec'¢; ={(2 -k2)V, ,2-2kV, , u., cot 4>-k”u..' cosec2¢>}/2g. (IO) Putting this in the expression for the efficiency, we find a considerable increase of efficiency. Thus with ¢=9O° and k=%, 1;=% nearly, ¢ a small angle and k = 5, 1;-= I nearly. With this arrangement of pump, therefore, the angle at the outer ends of the vanes is of comparatively little importance. A moderate angle of 30° or 40° may very well be adopted. The following numerical values of the velocity of the circumference of the pump have been obtained by taking k=%, and u., =o-254 (2gH). ¢> V. 90° -762/2gH 45; '842 H 30 '911 H 20° 1-023, , The quantity of water to be pumped by a centrifugal pump necessarily varies, and an adjustment for different quantities of water cannot easily be introduced. Hence it is that the average efficiency of umps of this kind is in practice less than the efficiencies given above. The advantage of a vortex chamber is also generally neg ected. The velocity in the supply and discharge pipes is also often made greater than is consistent with a high degree of efficiency! Velocities of 6 or 7 ft. per second in the discharge and suction pipes, when the lift is small, cause a very sensible waste of energy; 3 to 6 ft. would be much better. Centrifugal pumps of very large size have been constructed. Easton and Anderson made pumps for the North Sea canal in Holland to deliver each 670 tons of water per minute on a lift of 5 ft. The pump disks are 8 ft. diameter. ]. and H. Gwynne constructed some pumps for draining the Ferrarese Marshes, which together deliver 2000 tons per minute. A pump made under Professor ] Thoms0n's direction for drainage works in Barbados had a pump disk 16 ft. in diameter and a whirlpool chamber V32 ft. in diameter. The efhciency of centrifugal pumps wl1en delivering less or more than the normal quantity of water is discussed ina paper in the Proc. Inst. Civ. Eng. vol. 53. § 211. High Lift CentrUugal Pumps.-It has long been known that centrifugal pumps could be worked in series, each pump overcoming a part of the lift. This method has been perfected, and centrifugal pumps for very high lifts with great efficiency have been used by Sulzer and others. C. W. Darley (Proc. I nst. Civ. Eng., supplement to vol. 154, p. 156) has described some pumps of this new type driven by Parsons, steam turbines for the water supply of Sydney, N.S.W. Each pump was designed to deliver 1% million gallons per twenty-four hours against a head of 240 ft. at 3300 revs. per minute. Three pumps in series give therefore a lift of 720 ft. The pump consists of a central double sided impeller 12 in. diameter. The water entering at the bottom divides and enters the runner at each side through a bell-mouthed passage. The shaft is provided with ring and groove glands which on the suction side keep the air out and on the pressure side prevent leakage. Some water from the pressure side leaks through the glands, but beyond the first grooves it passes into a pocket and is returned to the suction side of the pump. For the glands on tho suction side water is supplied from a lowprcssure service. N0 packing is used in the glands. During the trials no water was seen at the glands. The following are the results of tests made at Newcastlez—I 0 9 I. II. III. IV. Duration of test hours 2 1 -54 1-2 1 - 55 Steam pressure lb per sq. in. 57 57 84 55 Weight of steam per water h.p. hour ..... Tb 27-93 30-67 28-83 27-89 Speed in revs. per min. . 3300 3330 3710 3340 Height of suction . ft. II II II II Total lift ... ft. 762 744 917 756 Million galls. per day pumped-By Venturi meter 1-573 1-499 1-689 1-503

By orifice .... 1-623 1-513 1-723 1-555

Water h.p. .. . 252 235 326 239 In trial IV. the steam was superheated 95° F. From other trials underthe same conditions as trial I. the Parsons turbine uses 15-6 lb of steam per brake h.p. hour, so that the combined efficiency of turbine and pumps is about 56%, a remarkably good result. § 212. Air-Lift Pumps.—An interesting and simple method of pumping by compressed air, invented by Dr ]. Pohlé of Arizona, is likely to be very useful in certain cases. Suppose a rising main placed in a deep bore hole in which there is a considerable depth of water. Air compressed to a sufficient pressure is conveyed by an air pipe and introduced at the lower end of the rising main. The air rising in the main diminishes the average density of the contents of the main, and their aggregate weight no longer balances the pressure at the lower end of the main due to its submersion. An up- ward How is set up, and if the air supply is sufficient the water in the rising main is lifted to any required height. The higher the lift above the level in the bore hole the deeper must be the point Air Suplpy

= 1.

Sbnd Pipe "'f'° Hn ro Rewm? ni; L fi: 2 it .. Ground ' L é, .C ., A UQ' 1 q* rv t

J 1: ii-:-ra ' -'. ~'?;'~. !E:: 'r Pg* ll K ', :Hn, Q I 2»<'“' ln- "=. <13 2:1-.§ =' ~. 3 -, Af ..~, if 4:.:, 1:1 "7»'Iv- 'Iv git., gf-, , I 'il i at which air is Z; . injected. , qi ff, ,I ', ¢ 4 212 shows an air- , ' Q 7 2.55 -'=' , lift pump con- """"“ fl "" " . 'Ef'" structed for W. - » gf!" T", § b;, f57gffx H. Maxwell at Air Pipe 2f'D/amthe Tunbridge ' 5; Wells water- iii! works. There is a """ two-stage steam air compressor, compressing air to ', FIG. 212. from Q0 to 100 lb per sq. in. The bore hole is 3 50 ft. deep, lined with steel pipes I 5 ini diameter for 200 ft. and with perforated pipes 13%, in. diameter for the lower 1 50 ft. The rest level of the water is 96 ft. from the ground-level, and the level when pumping 32,000 gallons per hour is 1 20 ft." from the ground-level. The rising main is 7 in. diameter, and is carried nearly to the bottom of the bore hole and to 20 ft. above the ground-level. The air pipe is 2% in. diameter. In a. trial run 31,402 gallons per hour were raised 133 ft. above the level in the well. Trials of the efficiency of the system' made at San Francisco with varying conditions will be found in a paper by E. A. Rix (laura. Amer. Assoc. Eng. Soc. vol. 2 5, 1900). Maxwell found the best results when the ratio of immersion to lift was 3 to 1 at the start and 2.2 to 1 at the end of the trial. In these conditions the efficiency was 37% calculated on the indicated h.p. of the steam-engine, and 46% calculated on the indicated work of the compressor. 2.7 volumes of free air were used to 1 of water lifted. The system is suitable for temporary purposes, especially as the quantity of water raised is much greater than could be pumped by any other system in a bore hole of a given size. It is useful for clearing a boring of sand and may be advantageously used permanently when a boring is in sand or gravel which cannot be kept out of the bore hole. The initial cost is small.

§ 213. Centrifugal Fans.—Centrifugal fans are constructed similarly to centrifugal pumps, and are used for compressing air to pressures not exceeding 10 to 15 in. of water-column. With this small variation of pressure the variation of volume and density of the air may be neglected without sensible error. The conditions of pressure and discharge for fans are generally less accurately known than in the case of pumps, and the design of fans is generally somewhat crude. They seldom have whirlpool chambers, though a large expanding outlet is provided in the case of the important Guibal fans used in mine ventilation.

It is usual to reckon the difference of pressure at the inlet and outlet of a fan in inches of water-column. One inch of water-column = 64.4 ft. of air at average atmospheric pressure = 5.2℔ per sq. ft.

Roughly the pressure-head produced in a fan without means of utilizing the kinetic energy of discharge would be v2/2g ft. of air, or 0.00024 v2 in. of water, where v is the velocity of the tips of the fan blades in feet per second. If d is the diameter of the fan and t the width at the external circumference, then πdt is the discharge area of the fan disk. If Q is the discharge in cub. ft. per sec., u = Q/π dt is the radial velocity of discharge which is numerically equal to the discharge per square foot of outlet in cubic feet per second. As both the losses in the fan and the work done are roughly proportional to u2 in fans of the same type, and are also proportional to the gauge pressure p, then if the losses are to be a constant percentage of the work done u may be taken proportional to √p. In ordinary cases u = about 22 √p. The width t of the fan is generally from 0.35 to 0.45d. Hence if Q is given, the diameter of the fan should be:—

For t = 0.35d,    d = 0.20 √ (Q / √p)
For t = 0.45d,    d = 0.18 √ (Q / √p)

If p is the pressure difference in the fan in inches of water, and N the revolutions of fan,

v = πdN/60 ft. per sec.
N = 1230 √ p/d revs. per min.

As the pressure difference is small, the work done in compressing the air is almost exactly 5.2pQ foot-pounds per second. Usually, however, the kinetic energy of the air in the discharge pipe is not inconsiderable compared with the work done in compression. If w is the velocity of the air where the discharge pressure is measured, the air carries away w2/2g foot-pounds per ℔ of air as kinetic energy. In Q cubic feet or 0.0807Q ℔ the kinetic energy is 0.00125 Qw2 foot-pounds per second.

The efficiency of fans is reckoned in two ways. If B.H.P. is the effective horse-power applied at the fan shaft, then the efficiency reckoned on the work of compression is

η = 5.2pQ / 550 B.H.P.

On the other hand, if the kinetic energy in the delivery pipe is taken as part of the useful work the efficiency is

η2 = (5.2 pQ + 0.00125 Qw2) / 550 B.H.P.

Although the theory above is a rough one it agrees sufficiently with experiment, with some merely numerical modifications.

An extremely interesting experimental investigation of the action of centrifugal fans has been made by H. Heenan and W. Gilbert (Proc. Inst. Civ. Eng. vol. 123, p. 272). The fans delivered through an air trunk in which different resistances could be obtained by introducing diaphragms with circular apertures of different sizes. Suppose a fan run at constant speed with different resistances and the compression pressure, discharge and brake horse-power measured. The results plot in such a diagram as is shown in fig. 213. The less the resistance to discharge, that is the larger the opening in the air trunk, the greater the quantity of air discharged at the given speed of the fan. On the other hand the compression pressure diminishes. The curve marked total gauge is the compression pressure + the velocity head in the discharge pipe, both in inches of water. This curve falls, but not nearly so much as the compression curve, when the resistance in the air trunk is diminished. The brake horse-power increases as the resistance is diminished because the volume of discharge increases very much. The curve marked efficiency is the efficiency calculated on the work of compression only. It is zero for no discharge, and zero also when there is no resistance and all the energy given to the air is carried away as kinetic energy. There is a discharge for which this efficiency is a maximum; it is about half the discharge which there is when there is no resistance and the delivery pipe is full open. The conditions of speed and discharge corresponding to the greatest efficiency of compression are those ordinarily taken as the best normal conditions of working. The curve marked total efficiency gives the efficiency calculated on the work of compression and kinetic energy of discharge. Messrs Gilbert and Heenan found the efficiencies of ordinary fans calculated on the compression to be 40 to 60% when working at about normal conditions.

EB1911 Hydraulics - Fig. 213.jpg
Fig. 213.

Taking some of Messrs Heenan and Gilbert’s results for ordinary fans in normal conditions, they have been found to agree fairly with the following approximate rules. Let pc be the compression pressure and q the volume discharged per second per square foot of outlet area of fan. Then the total gauge pressure due to pressure of compression and velocity of discharge is approximately: p = pc + 0.0004q2 in. of water, so that if pc is given, p can be found approximately. The pressure p depends on the circumferential speed v of the fan disk—

p = 0.00025 v2 in. of water
v = 63 √p ft. per sec.

The discharge per square foot of outlet of fan is—

q = 15 to 18 √p cub. ft. per sec.

The total discharge is

Q = π dtq = 47 to 56 dtp

For

t = .35d,   d = 0.22 to 0.25 √(Q / √p) ft.
t = .45d,   d = 0.20 to 0.22 √(Q / √p) ft.
N = 1203 √ p/d.

These approximate equations, which are derived purely from experiment, do not differ greatly from those obtained by the rough theory given above. The theory helps to explain the reason for the form of the empirical results.  (W. C. U.) 

  1. Except where other units are given, the units throughout this article are feet, pounds, pounds per sq. ft., feet per second.
  2. Journal de M. Liouville, t. xiii. (1868); Mémoires de l’Académie, des Sciences de l’Institut de France, t. xxiii., xxiv. (1877).